Installing anything creates a potential for vulnerabilities - e.g., if there's a vulnerability in the dropbox daemon (or whatever protocol you use for sharing files), or in any other software on your Linux box, it would be possible to exploit it (once it's known). You'll need to evaluate how much of a risk this is for each component that you want to use (e.g. an up-to-date vanilla SSH server install is somewhat safe for such home-server use that you're outlining - given other precautions)
For security considerations: I wouldn't use FTP, it's a plaintext protocol (which may be a vulnerability in itself: your password and your files go over the wire in the clear); I'd go for a SSH server - gives you shell access and SFTP (secure file transfer), both encrypted; it is possible to limit access only to SFTP without a shell though.
Also, there are various helper services which can help against automated attacks, e.g.
fail2ban (blocks an IP address from connecting after several failed attempts).
Usual other considerations apply: don't allow superuser (root) remote access, don't use weak passwords (and if possible, don't allow authentication by password at all, instead, use public key authentication and encrypt the private keys), update regularly (or set up automatic update), etc.
Also, block any access to the box except what is actually used (e.g. using
iptables, deny everything, then only allow what's needed) - this is a lesser concern on most Linux installs, as there aren't many services open by default, but still worth mentioning.