ASP.Net already provides a static instance of the current membership provider via the static Membership class and its static Provider property. The binding would likely be in your Application_Start method and look something like this:
Bind<MembershipProvider>()
.ToMethod(ctx => Membership.Provider);
Again, because the Memberhip.Provider is a static, it's sort of like a singleton already, so the behavior you try to apply doesn't really matter as much.
By not specifying any behavior in the above snippet, Ninject will default to a transient behavior. In this sort of a binding, I believe that will amount to calling the lambda that returns Membership.Provider every time it needs to inject a MembershipProvider type.
I suppose there could be an argument for explicitly specifying a singleton behavior as Ninject would likely "cache" the value returned by the lambda the first time it needs to inject a MembershipProvider, in effect saving the overhead of executing the lambda. I'm not 100% sure that's how Ninject would work in this situation, but it seems reasonable that it would.
All that said, my personal preference would be to use OnePerRequestBehavior, this way I know Ninject will call my lambda once for each request. Not sure it's necessary, but I like the idea of getting the provider from Membership.Provider once each request since I suppose you can't make assumptions about how or when Membership.Provider get's set, though you could probably find out if you dig enough with Reflector.
Bind<MembershipProvider>()
.ToMethod(ctx => Membership.Provider)
.Using<OnePerRequestBehavior>();
Good luck. Sorry your question sat out here so long!