If you only fly in the continental United States then the answer is no. If you're considering a flying adventure to/through Mexico/Canada or to the Bahamas, then the answer is yes. If you are a flight instructor I highly recommend you obtain one.
Who's going to check?
According to my clients, Customs and Border Agents are beginning to check FAA/FCC documents so it's becoming more important.
The FAA does not enforce FCC rules or vice-versa. I've never been asked in over 30-years. However, it might be your dumb bad luck that you run across a border agent (theirs or ours) who just broke up with their significant other and is looking to take it out on someone.
The instructions below assume you are a non-commercial operator and that you are flying a private aircraft.
First you must register with the FCC in the Universal Licensing System (ULS). They will assign you a number/password for you to access the system. The number is similar to your pilot certificate number and will stay with you for life. If you already have an FCC license use the FRN you already have.