As was quoted by a member in another list: "A Basenji is a slob's best friend." Meaning whatever you do not pick up and put away, the B has full permission to do what they do.
I try to always have two barriers between them and freedom. There are always "slips" with a single gate. In the house we have two childproof (?) gates, that the dogs respect, but occasionally when we are slack, they will open the gate and allow themselves into the kitchen (a banned place for them, and me!).
The boundary fence if it looks a challenge, the B will take the challenge and try to defeat it - by whatever means. If they decide to climb, a couple of strands of saggy wire towards the top will most usually defeat their attempts. Especially if you can put some of the stays cranked inwards, so that by the time they get to the wire, it will sag, and they will fall back into the shrubbery which after a couple of times will pass the message onto them that this is not the way!
As for digging, lay some reo wire mesh flat on the ground, with squares about 75-100 mm (3 to 4 inches) spacing. Use any reo that you can find and fix it with tent pegs or similar, to keep it stationary. This should be placed close to the fence as possible, so that the lawn mower can still control the grass.
Do not expect miracles at first, but mine are left alone while we go shopping, (and the regular shops are 40kms (about 25 miles) away), so it is never for 10 mins. They now know to behave, as we usually come back and they get a treat. Today was raw beef brisket bones (RBBB). they love these, and they leave nothing behind for other insects, animals, or motor mowers to use as missiles to throw at the windows of the house. Do not use the "marrow bones" as they can crack and damage the teeth unless you want a closer relationship with your vet. These are the thick thigh bones of the cattle, and if you weigh about a tonne (ton), you would want strong bones to hold you up. The RBBB is the breast bone, and comes at about 1metre (3 feet) long, but the butchers usually discard them, so get to know when they are available, and shop for them.
That is enough for now, but feel free to ask questions about your young B. They are 2yo children, that want to be loved as part of the family.