The RSA key pair (actually, the secret key) is not tied to the specific computer. (It may be tied to the IP address you are connecting from, but this is reasonably rare in common setups.) So you shouldn't need to do anything specific simply because you got a new computer.
However, you may want to take this opportunity to generate a new key pair. Good key hygiene involves regularly replacing cryptographic keys, particularly long-term/high-value keys such as SSH authentication key pairs. Depending on how your SSH environment is set up, you may need to coordinate this with the system administrator. I personally like to replace long-term keys every one to two years unless simply the act of replacing the key causes significant inconvenience.
It seems highly likely that your SSH (or SFTP, or SCP; they are all the same as far as this is concerned) client is not using the key pair to attempt authentication. For that reason, public key authentication would be unavailable, and if the server is configured to only accept public key authentication then this will cause authentication to fail, further leading to the connection attempt failing.
Transferring the key pair from one system to another is generally a two-step process:
- Copy the relevant files, making sure that you copy both the public and private key
- Point the SSH client on the new system to the private key file on the new system
While the exact steps will obviously vary depending on your client software, look for settings relating to authentication, key selection, and similar. Point the client software to the key file you copied from your old system, and attempt to connect. If you have configured your client correctly, it should use the key pair to authenticate, which will allow you to log in.
Look at any connection settings dialogs or similar to determine what any differences may be. Ideally, they should look identical (except for possible differences relating to different software versions).