I believe socio-economic status does not have to be a factor in relationships. However, as you've found, it does get awkward at times. I suggest that with time you and your new friend will work out a way of relating.
I suggest she is disappointed that she cannot afford a gym membership but that doesn't mean that she will be less of a friend or that you'll be less of a friend because of it.
I've had friendship cross socio-economic boundaries as well as other boundaries such as they're married or have children, are in school, or in a different occupation. What matters is whether or not you enjoy each others company. I does take time to work out the ways in which you're comfortable with each other.
You're feeling the awkwardness of a new friendship in which you really don't know what she was thinking. This gives you an opportunity to talk about your differences as well as similarities and how to manage them.
Having this conversation can make you closer to each other or can show that the friendship won't work.
Please do not feel guilty. What do have to feel guilty about? That you have more money than she has? Should you then give away your money so that you can be equal money wise? Should you not join because she can't? I say no. Find the other things that you have in common and focus on those with each other.