Just pick. It doesn't even matter if the stem stays on the plant. If any pruning, do remove a few of the developing green fruits on larger clusters of larger fruited types or severely misshaped green tomatoes. I let a few Mr. Ugly tomatoes ripen with larger blossom scars and now I will have to eat them rather than sell them. No harm to the plant but small lost $.

This is from the About.com: Home Cooking page on Tomato Storage and Selection Information:

"Refrigeration is the enemy of the tomato as it nullifies flavor and turns the flesh mealy. The culprit is a compound called Z-3 hexenel, which accounts for the tomato's scent and taste. The development process which turns tomato's linolenic acid to the Z-3 that makes our mouth and nose sing is hindered by cold. If you must refrigerate a tomato, take it out about an hour before using it to let it return to room temperature to revive any lurking Z-3."

I hope that helps.

Betsy

Here is a link that might be useful: 5 Common Mistakes that Ruin Your Fresh Tomatoes, and How to Avoid Them

Betsy,I am so happy you posted that link. If you ask 100 people on the street why super market tomatoes are tastless 90 will say "because the are produced in hothouses." The information also tells use why saving seed from a beautiul store bought tomatoe is a bad idea.
And for those who have always known the 5 tips,it's comforting to see from the pictures that home grown tomatoes arn't supposed to have unblemished skin. Kinda like a lady with perfect skin ,soft hands and long manicured nails. You know she doesn't wash dishes nor garden so that's 2 strikes starting off.