Should I use RSA encryption since RSA is said to be broken by NSA? I am developing a simple web app that uses asymmetric encryption, but I am not sure whether to chose RSA for the reasons mentioned above. 

What are your suggestions? Also if I decided to go with RSA what is the recommended Key Size ?

Also, someone in the comment area mentioned using libraries made by others, but did not say which libraries should I use?
This question has many problems in the way it was asked, and clearly did not come after doing some investigation. However, since this seems to be a misconception that is spreading widely, I will relate to it.

It is not true that the "crypto community" (whoever that is) believes that the NSA can break RSA. In fact, if Snowden taught us anything, it is that the NSA is using many techniques to bypass RSA in TLS and elsewhere (stealing private keys, utilizing implementation bugs, and more), but are not breaking RSA. If they could break RSA then this wouldn't be necessary and would be the easiest route to take. 

Now, it may very well be that the NSA has some advantages in factoring over the academic and open crypto community. This makes a lot of sense for two reasons:


We share everything that we discover with them (by publishing papers), and they share nothing about what they discover with us
They have absurd computing power that they can utilize


Having said the above, I do not think that they can do much more than the rest of us. What's my basis for this belief? Nothing except that many very strong academic minds have also worked on this, and the Snowden revelations about what they are doing.

This possibly justifies the move to RSA-2048 (instead of settling on RSA-1536). I am very skeptical that the NSA can go beyond RSA-1024 by much. However, I think that it is a good idea to use RSA-2048 because I strongly believe that this is way beyond their power, again for the reasons stated above.