Essentially then, you have the same idea you have today: those who will rely on UBI are the same people who rely on social benefits today, and those who do not get it are the ones who will have to fund it (with a slightly enlarged soft border between the two). The stigma isn't because people know that you get social benefits (the records aren't public), it's that they don't see you working.
The cut-offs are an issue, but we already have Mini (up to 450€) and Midi Jobs (up to 1300€) where you'll not get any reduction in benefits for 100€ and 20% of whatever you make on top will also not affect your benefits. That's not a perfectly soft threshold, but it's not a hard cut-off either. The thresholds that you lose money at are much higher (e.g. earning just a little bit too much to be eligible for subsidized housing, which is close to the median salary in most states).
1. a UBI would very likely be unconditional and universal (as long as you are a citizen) so relying on it would carry no social stigma, because everybody gets it.
2. you still get a UBI if you get yourself a job – which means a job gives you extra money. With most current unemployment benefits you could sometimes loose money when you get a job, or you'd just get a tiny bit extra for a lot more work (especially if you are doing demanding physical labour).
This means the incentives are completely and utterly different – and incentives kinda matter when we want to discuss human behaviour.