The RSA algorithm is not necessarily out of date, but the key size is important. It is still used regularly all over the place. An RSA key length of 3072 bits should be used if security is required beyond 2030. This is just a function of the improvements in computing power and not a weakness in the algorithm.
I was looking at the LockBox prior to SongBeamer starting to maintain it (Delphi 2007) I found the RSA unstable - just crashing or producing wrong results from time to time. It may have been fixed since then though. If you are going to use it, make sure it supports the key lengths you need, and then create some automated tests with known inputs and outputs. Make sure it runs stable. If it does, then use it by all means.
You can also use the Microsoft Crypto API to do your RSA encryption. I wrote a wrapper for Delphi 2007, but have not updated it for Unicode (I know, shame on me). Someone else might be able to update it to work with Unicode easily.
Another option would be to use the .NET Framework cryptography functions via COM interoperability or something.