I was 6'5" and was able to play 5 years professionally with the Red Sox and Devil Rays. The only thing that held me back from getting to the bigs and playing every day was hitting that da** slider. Defensively, it was an asset my pitchers loved because they had a huge target to throw to and it was easier on my body wear and tear wise because I had more size and didn't get tired as easy...not to mention helping out in the collisions at the plate.
I will tell you that you will have to work twice as hard on keeping your foot speed as quick as the smaller guys, but if you are wanting to climb that ladder as far as you can go, you aren't afraid of a little hard work.
You would be surprised how many "big" (6'2+) catchers there are now a days in Division I and professional baseball. Just work at all the small things and you will be fine.