A static IP is usually just as secure as a dynamic IP, though a lot depends on:
- How 'dynamic' is the 'dynamic IP`. Over here (NL) a dynamic IP often stays the same for many months, or even years. In that case the difference is moot.
- Are you using a proxy or a VPN? Is not your IP will show up in all communications with the outside world. Both if you have a dynamic IP and if you have a static IP. The only difference will be that it might easier to find who has a fixed IP at a specific time rather than who had a dynamic IP. (For the latter they would need top check the ISPs logfiles).
As to having a DNS name pointing at you: This is not really related to a fixed IP or not. It is true that most people who go though the effort of getting -a now rare- static IP also go though the effort of getting a DNS name. But there is nothing stopping you from getting a name for a dynamic IP (e.g. via DynDNS and similar services).
Edit: I seem to have skipped the “What about privacy concerns” part.
To answer that: There are often better ways to identify a person than using the IP address.
Since an IP can be used by the whole family, guests etc etc. It does not point to a single person.
A single computer/browser on the other hand usually points to a specific person and identifying and tracking that person via cookies, via information stored via flash (aka supercookies) or via browser plugins (checking and remembering which ones are installed. Not via a trojan horse type plugin) is often a much bigger danger.