I finally figured these things out by child #3...
It's usually more a problem with the MOM being a worry wart than the child freaking out that they aren't with you
Even when they do cry, it's because of fear of the unknown. If they haven't been away from you then they don't understand that you actually WILL be back soon-- and more importantly, that they will be FINE without you for a brief amount of time
Once they get over the initial shock of being with someone besides you, they appreciate the time without you - seriously, they love you as much as you love them but we all need a break sometimes - including that they need a break from you!
Once they get used to being in the new environment, they will LOVE, LOVE, LOVE exploring new toys and being around other children. Even the shyest of kids eventually get used to other kids being in their space, and even if they don't interact they enjoy seeing humans their size and will stare at them and watch them play
It's very healthy for both of you!!!
Since you've already paid for the membership, what do you have to lose??? Can you go as often as you want? Try going several days in a row for VERY small amounts of time (seriously, start with like 10 minutes so she can see that you ARE going to come back and that the world did not implode while you were gone) and increasingly leave her for longer peroids of time until you are able to leave her long enough to get a good workout-- and trust me if you stay consistant, she will ASK to go!!
As a parent and former child care worker, I can tell you the worst thing to do is linger when you drop her off. You should go by there one time to tour the child care center. Stay there with her and walk around with her. Don't let her touch anything yet. See if she would like to stay and play, but let her know mommy can't stay there and will be right back. If she doesn't seem OK with the idea, let her play VERY briefly and then take her home. Take note of something special she really liked and talk about it the whole way home. Ask her if she'd like to go back some time to play with it again. Try and head back within the next day or 2 and talk about it all morning to build up her excitment. Then once you get there, do NOT linger for a long time and try to be conscious of your facial expression-- don't look worried-- smile at her and tell her you'll be right back-- then hug her once and walk out! Go back in 10 minutes to check on her. If she is OK, stay away longer. If not, go in and ask if she wants to go home or stay and play. If she wants to stay and play, leave again for another 10-15 minutes and so on. You can do this!!