our boys are both BRAT dogs. they came "trained" to go on demand. that helps a lot. they both HATE the rain, or drizzle, or whatever…. if on demand fails, they will both go if i walk them. i bought a $175 rain/waterproof suit i wear and a BIG golf umbrella. I dont get very wet. they do. but if we walk about 2 blocks they decide to go and get back in. except in lightning. i HATE lightning.
I got to the point where I would just hook Paco up on an extendible leash on the balcony where we had him going, and closed the door. I'd watch him through the window while I got work done and the MOMENT he peed I'd open the door, call him in, praise him, give him a treat, etc.
Now, if I put him on the extendible, he automatically goes to the bathroom because he thinks that's how he'll get taken off. 
You just have to walk a bit. I'm pretty lucky as Buddy doesn't mind the cold and rain and will happily go for a stroll and get soaked or just run outside and pee.
I take 1 dog at a time, with a big umbrella I hold over THE DOG, and take them to a favorite bush (for Eddie) and the end of the driveway (for Nicky), and as soon as they pee, we race back inside. And they are 11 and 13 years old.
Now, neither will soil the house, they will eventually go out the dog door into the back yard, but doing this umbrella dance a couple of times a day makes them very happy in foul weather. Dexter will likely never want to be out in the rain, so keep him as dry as possible, get him moving (do the walk when it is not raining, so he is used to walking a bit to pee, then it will be easier when it's raining).
Pippin doens't like it either but will after a while..if my husband goes with him (he gets the late night shift with the pack..I do the early morning), he resort to pee on my front porch..my hubby doesn't see it so he gets away with it..but my outdoor rug has a million round circles on it so I know he does it…
My other dogs, non b's could not care less if it rains..makes for more mud=happy dogs..
None of mine will go outside in the rain unless I 'force' them to. Even the puppy has taken to this. In other words, I get to stand outside with them, until thoroughly drenched, then when they see that I won't let them come in side, no matter how wet I am, they will give a huff, snort, whine and then go and potty. Johnny will leap over the areas he considers untouchable to get to his particular potty spot. This goes for the Basenjified Boxer as well. She looks to see if it's raining, and if it is, she turns the other way. The only thing is, she figured out that the sooner she goes, the less time she has to spend outside in the rain!
I think the thing is either take them for a walk or go out with them - and don't give in until they go. I've always just stayed out with mine so they've understood that they won't be allowed back in until they do their business.
My two oldest would never dream of peeing in the house. If it is pouring, they will hold it and nap and never move - because they know the minute they move they are going outside (with me) and won't be allowed in until they go.
The 2 youngest don't even seem to notice that it is raining. Aaliyah was indoctrinated that she goes outside (no matter the weather) the first day she came home at 11 weeks. Consequently she accepts whatever the weather is and chooses to pee out in the open, even if it is raining. Ozzy simply has followed her lead and will also go to the center of yard in the rain. Ozzy and Liyah actually like to go out and play in the rain so they are a breeze.
Oh and all 4 of mine take walks in the rain (unless it is a deluge).
We have two different tactics for getting Dolce to go in the rain. The first is to physically throw her out into the middle of the yard. Once she realizes her feet are already wet and they are going to get wetter trying to make her way back inside she must think well my feet are already wet I may as well empty my bladder. The other is to parade her around on the leash in the backyard until she goes. Though, I much prefer the first tactic so I dont have to get my feet wet.
We have grassy dog run on the side of our house- that is his usual potty area. If it has been raining, he will not go near the wet grass! Rather, he potties on the back deck- every single time!!! Totally drives us nuts!
The only thing is, she figured out that the sooner she goes, the less time she has to spend outside in the rain!
I pray for that day to arrive. 
I insist that mine pee even if it's raining but they'll stand for ages in the pouring rain hoping that I'll give way. No thought of peeing quickly and getting back in the dry, just stubbornly hoping that they can outwit me!
Lycia has a bladder of steel.. I have seen her go almost a full 24 hours with no peeing due to adverse weather. hehehe
When its just a light rain mine will crawl under the deck and go but if it's a downpour, then I have to pick her up and carry her down the block and make her walk back. She definitely wants to get home when I put her down, but the running to get home makes her want to pee. Mission accomplished. Be sure to treat big when she does it.
As they say, the key is do not come in till he goes, period. And you might want to consider walking him a bit… helps to encourage them to go potty
when we first got Kenji he didn't like the rain at all. He did the same thing. But I found that taking him out soon and offten in the rain got him comfortable in the rain… He still dosent voluntarily walk into mud or poddles or the rain but he realizes that it won't kill him.
first I started taking him out and if he doesn't pee he goes right back in his crate. After 15 to 30min I take him out and straight to the door. Ring bell and "hurry up potty" again. If he pees and or poops, praise, go right inside and he gets to play for a while. Also start walking him in the rain. When I walked kenji in the rain he would pull for a while and walk between my legs but keep walking and try to keep him to your side. Again this is what worked for me but whatever you do, do it all the time and consistently.
Lycia has a bladder of steel.. I have seen her go almost a full 24 hours with no peeing due to adverse weather. hehehe
LOL, Cody too, although if it had been up to him, he would have waited even longer!
When it's raining, he'll stand at the door for a few seconds like he's weighing his options. "Should I get wet and dirty, or should I just hold it?" When it's a heavy storm, he doesn't even pause, he just attempts to run back into the house. On those days I have to pick him up, carry him into the back yard to his spot, wait for him to do his business, and then carry him back inside. If I didn't do that, he would just stand there like he was paralyzed by the rain (his own personal kryptonite!). He has me trained well. Luckily we don't get many storms like that 
We live on the coast in BC, so Kipawa, who was born in the Pacific Northwest, has realized rain is a part of life here. Of course he prefers dry weather (what basenji wouldn't!), but if he really has to go, he will go in any weather. Certainly it does help if I am there, also getting wet, saying "hurry hurry". I believe that because he also sees me getting wet, he figures that the situation is then fair.
When it is raining, I have to take the dogs out 1-2 at a time and stand out in the rain with them until they go.
Well, it was storming outside, and I stood in the rain with him for about 7 minutes, but all he did was cry horrendously, run around in circles, pull on his leash, and try to hide under my legs to avoid the rain. 7 minutes in a downpour is a lot to me…lol. It just didn't seem like standing and waiting would have worked. I'm pretty sure I would have had to stand till he uncontrollably peed on himself lol, and who knows how long that would have taken. I still need to get an umbrella /sigh
Walking helps to stimulate the bladder. Usually if they have to go they will hurry to do their business if they know they will get to come back in. My boy will run out to the tree pee as fast as he can and then pull to get back in but that is because he learned if he doesn't we will walk until he does.
