The closest airport is the Newark International Airport. For those coming from Europe, JFK might be the only possibility; it is about a 2 hour drive from Princeton. Philadelphia Airport is also a possibility; it is about a one-hour drive away.
If you arrive at JFK or Philadelphia, you probably should rent a car. From Newark Airport, you can either rent a car and drive, or take the Princeton Airporter. The Princeton Airporter serves JFK as well, but less frequently.
To get here from Newark Airport, first follow the signs for New Jersey Turnpike South, get on it going south (toward Trenton). Get off at Exit 9, New Brunswick. Immediately after the toll booth, turn right and take Route 18 North (that's correct, North!). Move to the second lane from the left right away, and watch for the sign for Route 1 South. Exit there, take Route 1 South, all the way to Princeton (about 20 minutes). Pass the Princeton exits labeled Harrison and Fisher, then exit and turn right to Washington Road. Now, start counting lights. At the third light, turn left, you will be on Nassau Street. Pass the first light, and turn right after one block (there is a kiosk at the corner) - Nassau Inn is the big building facing you. Go half a block, turn right, curve left, and stop at the entrance (under the arch) to the Inn. I think the whole thing will take about 2 to 2 1/2 hours after you land at the Airport.
From the JFK Airport, follow the signs for Belt Parkway, take it going East. Exit it to take the Verrazona Bridge, pass it, continue East on 287, follow the directions for New Jersey Turnpike, take it going South. Follow the instructions above for Newark Airport.
From Philadelphia Airport, take US 95 North. Remain on it for nearly 45 minutes. After you enter New Jersey, pass the exit for 206, and exit at the next chance to take Princeton Pike North. Go all the way to the end (about 15 minutes), you will come to a T-stop at Nassau Street. Turn right, go two short blocks, turn left at Palmer Square (there is a kiosk there) Nassau Inn is the big building facing you. Go one block, turn right, follow the road as it curves left, stop under the archway, in front of Nassau Inn.