I have basically zero meditation training, so I don't know what I'm doing, but I also experience this. Whenever I really quiet my mind and allow myself to notice thoughts and feelings non-judgmentally, I spend a portion of basically every session in tears.
It usually happens when I start noting raw emotions themselves. Sometimes they have an accompanying thought, but often they don't. I suspect it's because as I go through my day, I rarely have the space to express my emotions, so I just avoid/suppress them without resolving, and they all come roaring up when I allow myself to connect with them again.
Personally, I always feel significantly better after this, so I see no reason to stop, but I find it a little odd that the primary thing I ever get out meditation is a big cry. It's good to know I'm not alone.
Is your crying cathartic like that? Do you feel better afterwords? Perhaps it's not a problem that you are allowing yourself to feel it.