Your coworkers are right—go ahead and try courting her. If you like her, someone else might get ahead of you, and you could end up regretting it later.
You’re a man, so be confident and take the lead. Invite her out on a date. Who knows, maybe she actually likes you too and is just waiting for you to make a move.
If you keep hesitating, she might start thinking her effort is wasted and that you’re not interested in her, which could make her feelings fade if she has any for you at all.