Well, I think the first thing you should ask yourself is, "so what?" I have news for you: Almost everyone is bi-curious, whether they want to admit it or not.
Now that we got that out of the way, it’s time to really address the issue. If your significant other has matured enough to realize that they might be bisexual, then listen to them openly. How you react to their potential "coming out" may impact them for the rest of their lives. Handle this like you would handle any other thing in your relationship. Be attentive and respectful.
Now, I know that this can be easier said than done. Your primal instincts might kick in, and you might feel hurt, jealous, or weirded out. This might be harder for girls to deal with than guys. I know it’s a double standard, but more guys than girls will probably be cool with their significant other being bi-curious. Guys might think, "sweet, this chick is awesome" while girls might think… well they will probably think it’s strange. More than likely, they will suspect that their boyfriend is gay, and then they will not be interested in them anymore. The relationship might become awkward and you may not know where things stand anymore.
However, if you really love the person that you’re with, then you should listen to them. If it’s too much for you to handle, then tell them that while you’re still there for them, they should probably talk to someone else about it, and start seeing other people. After all, you are just human, and it’s nearly impossible not to be extremely biased in this situation. When it comes down to it, you have to do what you’re comfortable with, just like every other situation you might have to deal with while in a relationship.
But, if it’s a semi-new relationship and this person is just realizing that they’re bi-curious, then the road they might be traveling will probably be long and hard (no pun intended). I know it sounds shallow, but we’re young and this is the time in our lives when we’re supposed to be free and happy. So, before you make any rash decisions, think seriously about whether you are committed enough to deal with this right now.