We're on Basenji #4 and don't worry, it gets MUCH easier very soon. It won't be long and Roxy will probably be able to roam the house while you're gone and fulfill the portrait of the family dog (not being facetious). We had 2 which did for 10 years and also had early problems.
Our first, Brie, was diagnosed with separation anxiety. We got her at 16 months old. Do a search online and you can find great tips if that is in fact the problem. Start with small steps – teach Roxy to sit and stay, then stay while you leave the room, then you can start to leave the room and close the door but come back in a few seconds, then a lengthen it, etc. Teach her that when you leave, you WILL come back. The biggest tip I found (and difficult for us) is to make leaving and coming home totally uneventful. Put Roxy in her create with a treat (or several) but no praise (that only will get her excited). Same, and especially, when you get home. Ignore her. Don't look directly at her but get her outside right away (you may have to go out with her to get her to do her business), THEN praise her when she does... now you can act normally -- just not at first. That is the advice our vet gave and helped. Anxiety and excitement feed off of eachoher. If possible, walk her before crate time.
Determining whether Roxy has separation issues or crate issues may also determine how soon she can be trusted alone -- yes, Basenjis can be trusted! At first Brie would destroy our house if left alone, but she had issues. After months of working with her we could keep loose in a bedroom (with a view outside) during the day (she did better than being in a crate). When we got #2, Ramsey, we had him and her in their bedroom (he in a crate at first) and by 2 we let him and Brie have full house access while we were gone from 8-5:30... and that was 10 years ago. Rarely a problem, but they also had each other.
Now I have two 11 mo olds -- Bonnie and Smitty. We got Bonnie when she was 4 mos (after loosing Brie) and she was quickly crate trained. She'd even slept in it during the day with the door open. BUT I wouldn't trust her loose if we were gone. She gets bored and chews. By contrast, Smitty HATES his crate. We got him just 6 weeks ago at 9 mos. He doesn't have separation anxiety. He can be in another room without us or hang outside. But he seems to have a phobia of confinement -- latch the crate and he freaks out. We're trying to make his crate a "happy place" but doing the above but giving him a pig ear to chew on to make him forget the door is closed (avoid raw hide when you leave) -- I have a post on the topic. We gave up on Smitty sleeping in his crate at night because we found out he really never slept much. He and the other two now sleep in our bedrooms, but the young ones don't get to leave the rooms at night, but sleep soundly ALL night (in our bed). I actually think Smitty may do ok left out, just not with our other 12 year old male!
I guess my point is there is no definitive answer, BUT there is hope. Once our house is down to 1 male, we'll start to leave Smitty out, then if all goes well, try leaving Bonnie out too.
Don't worry about starting over with the house breaking. They learn very quickly. With Bonnie we thought she'd never get going outside = potty time. I think it took about a month for her to start to get it then another of it being our fault (not catching the signs). You got to watch them like a hawk! Smitty was in a kennel and only took 3 weeks of watching the others to catch on. We still have accidents though, especially when its wet out. Like Duke said, don't become too confident thinking Roxy is trained completely -- let up your guard and it's a step backwards (or in a wet spot).