Typically, those routers also support access to the USB storage using FTP and/or SMB (Windows file sharing). In order to gain access externally, you'd just need to open the ports relevant to those services and port forward to the router's IP address (come to think of it, you may not even need to port forward since the public IP does belongs to the router).
As far as security implications, these protocols are only secured by authentication, and not the strongest authentication either. The data always travels “in the clear”. What you really want is something like SSL, where you can access those services in a completely secure fashion. That's why I wouldn't recommend opening the ports to these protocols on the router. Even the FTP and SMB implementations on the router are not necessarily invulnerable to attack.
What I would do is either use remote desktop solutions like LogMeIn or TeamViewer, or LogMeIn Hamachi. Now granted these require a running PC (and that may be counterproductive if your intent was to avoid running a PC by using the router as a NAS), but this makes it possible to completely secure your FTP/SMB sessions over SSL. And there are no ports to open or manage on your firewall.
The only other suggestion might be to use a dd-wrt/tomato compatible router and run a PPTP or OpenVPN server (on the router). PPTP isn’t necessarily the most secure protocol, but it does offer reasonable protection, and is simple to setup. OpenVPN is more secure, but more complex. Either can be problematic at times since ISPs often block their ports, or don’t support some of the underlying protocols (e.g., PPTP requires support for protocol GRE 47, many older routers don’t support it).