In most cases you can use CuteFTP from behind a router with no issues.
If you are having problems, try doing all your transfers in PASV mode.
Check with the FTP server administrator to make certain the Server and it's network are configured to allow FTP in PASV mode.
Check your router documentation for information on how to set it up for FTP, or how to open the appropriate ports. FTP generally uses port 21.
You can define a port range for CuteFTP to operate with under Tools > Global Options > Transfer Settings > Limit local port range from n to n.
You can try using UPnP (if your NAT or router supports it) to establish a connection properly. You can turn on port mapping under the Tools > Global Options > Connection tab.
To perform transfers in PORT mode, or for secure transfers, you will need to refer to your router's documentation for information on how to configure the router for such transfers.
If you are behind a NAT firewall, and the Secure FTP server is on a different network, you will only be able to make data transfers in PORT mode using a Clear Command Channel SSL connection.