I make sure to date every page in my sketchbooks, so I can usually nail down an exact date for the first drawing I've made of a particular character. So I'll use that date for their birthday! (For characters I first designed digitally I use the last date modified (ie when the file was finished), or posted online since that's usually the same day.) I don't have to think too hard this way, unlike for details like names. Names are much harder for me XD
If I don't have a clear date for when a character was first designed, it's kind of up in the air. Sometimes I'll pick the day I named them (as I just hinted earlier, sometimes it can take me a while to actually name characters). Other times I'll be pragmatic about it and simply pick that very date I decided to give them a birthday as their birthday (haha). If inspiration strikes me I'll pick something else that feels significant, such as the day of a holiday or the birthday of some celebrity or historical figure I feel is relevant to that character. I've seen people say already that they decide birthdays based on what astrological sign they want their character to have, but that is a method I've never used, since I don't pay attention to astrology in relation to my characters, but it's an interesting idea! I can appreciate the convenience for people that are more into astrology than I am.
At the end of the day, it's more of a trivia thing for me and doesn't really have a bearing on the character's backstory or anything like that, and since my characters are static and don't age in real time that's probably all it'll ever be. I mean, if and when I get around to actually USING these characters for something, it might end up being relevant, but I'll cross that bridge when I get to it. Heck, for some characters it wouldn't even make sense to have a birthday on our own calendar system (example: my fantrolls, since Homestuck trolls canonically measure age/time in 'sweeps' that have a different length than our years). But I do it anyway because it's one of those 'fun fact' things I simply enjoy adding, and it can be a satisfying bit of flavor for characters that don't otherwise have a lot of info yet.