According to p. 187 of the PHB, you can use your Performance skill to maintain a wealthy lifestyle during downtime, if you have proficiency in it.
Does having the Jack of All Trades feature count?
For reference, the PHB states on p. 57 that
[...] you can add half your proficiency bonus [...] to any ability check [...]
It states "add half your proficiency" rather than "are half proficient," however that may be because it applies to things that you cannot be proficient in, e.g. initiative.
No, Jack of All Trades doesn't count as having proficiency. All it does is change how you calculate your bonuses when rolling for a skill, and the rule for practicing a profession during downtime don't care about that. This rule only cares whether you really have proficiency in a skill (PHB, p. 187, emphasis mine):
If you have proficiency in the Performance skill and put your performance skill to use during your downtime, you earn enough to support a wealthy lifestyle instead.
So no, Jack of All Trades won't let you be a rock star. (On the other hand, what is a bard doing with her life if she doesn't have proficiency in Performance so she can be a rock star?!)
Jack of All Trades does not give you proficiency in any skill that you are not proficient in. It may grant you a bonus to ability checks that are otherwise made without proficiency, but it does not make you proficient. You are only proficient in a skill if you have a game feature which states that you are proficient.
