There are three factors that bear on this:
- Deciding whether to post.
- Separation of accounts.
- How you post.
Deciding whether to post:
Your question might already be answered, so be sure to search. If you don't find it, you need to decide how important it is to ask it. Could it reveal any private information? Do you risk looking deficient in some way in front of your coworkers?
I use my real name online and I've asked questions arising from current employment. However, they're general questions and I like to think I put some thought into them before asking. If my manager were to come across one of my questions, I hope the reaction would be "good question" or perhaps just "oh hey, Monica's there".
But sometimes you really do want to ask a question that you'd rather your manager and coworkers not know about -- you're having trouble dealing with that obnoxious coworker, that clueless team lead is messing up your deliverables, that dictatorial boss is demanding 80 hours a week for months on end... In those cases, read on.
Separation of accounts:
You're allowed to have more than one account on Stack Exchange so long as you don't use them to do anything you couldn't do with a single account (like vote for yourself, or double-vote on others' posts). The usual way to do this is to create a separate email address and OpenID credential and use the second account from a different browser or an incognito browser window. Basically, don't let the two accounts "meet" each other in the same browser session.
Note that two accounts means two separate inboxes; when you're logged in to your main account you won't see pings from comments on posts from the secondary account. It's more work to manage two accounts because you have to remember to check in. And you also need to keep track of what you did where; the first time you post from one account referring to "my previous question about X" that was actually from the other account, you've blown it.
Even if you do all that, though, you can still give yourself away by...
How you post:
People have individual quirks. The more-unusual ones, like idioms or turns of phrase that very few people use, might stand out to people who know you and recognize them. So might that word that you always misspell. It doesn't take a Big Data Extravaganza to find correlations in longer text samples.
You are probably a poor judge of how other people read what you write -- not because there's anything wrong with you, but because we all have blinders. You can reduce the impact by being aware of what you say and how you say it, but never, ever assume that you are truly anonymous on the Internet.
Knowing that, you can still ask your question -- and still add a layer of insulation by asking it from a second account. But don't ask about that jerk boss or that pompous coworker; ask your question neutrally and focus on yourself, the problems you are having. You and your boss disagree about priorities; your team lead is focusing on short-term deadlines and you worry about the long-term implications; you really really need to find a way to get a nap in the middle of the afternoon without people thinking you're a slacker; whatever. If you focus on you, then even if your coworkers do figure out it's you, they're more likely to treat it professionally and not take it personally.