Check the validity of a rsa key pair with only the public key?
While there are some obvious checks you can do, you can't cover everything:


You can check that the modulus is a composite odd number of the appropriate size
If you want to put in the effort, you can do a quick check if the modulus has any small factors
You can check if the public exponent is an odd number > 1


However, you can't check beyond that; you can't check if the modulus might not be specially built to be easy to factor, and you can't check if the public exponent is actually relatively prime to $\phi(N)$.  What is likely more important to you, RSA doesn't provide a way to verify that the public key entered is the specific public key of the intended receiver.

BTW: you said that this is a big security risk; what is the scenario that you fear that an attacker might introduce a bogus key, and gain some advantage?  If it's the fear that someone might substitute their public key, well, since they're likely to substitute a valid key, a validity check on the key isn't likely to protect you.  Instead, you need some other mechanism to tie the public key with the identity.