I wouldn't have been offended, and I'm Caucasian. "White people's problem" translates to a problem only people of means and privilege might have of any color. It's more appropriately called rich people's problems.
It's analogous to the phrase "first world hunger" in the facetious sense, as in 'I'm hungry but not really. I just want something to eat because I'm bored.' My wife and I use the phrase, and it's self-deprecating, not bigoted.
THIS, from Seinfeld, is first-world hunger:
Elaine: Come on. Let's go do something. I don't want to just sit around here. Wanna get something to eat?
Jerry: Where?
Elaine: I don't care. I'm not hungry.
Back to white-people problems, here's a wealthy fictional band member discussing his new spin-off ban. He doesn't use the phrase, but the meaning is the same - it's good to have rich people's problems:
William Murderface: You know what my fear is? I just hope this doesn't become bigger than Dethklok. You know, that's, that'd be... that'd be something, you know.
Dick Knubbler: Well, uh, I will tell you this, that's a good problem to have. That's a problem you want to have. It's a good one.