I assume you have tried multiple cables. If you do not have another one on hand to try, it's always good to have a spare!
If the problem is not your cable: clean out the port very well. Use a light and magnifier to make sure you get any visible debris. Then use a paper towel saturated with rubbing alcohol and some kind of thin non-metal tool to wipe the contacts on both sides of the connector.
If that doesn't do it, turn off the phone, remove your sim tray, and put the phone in a warm spot for a few hours.
If it still won't work, go to Home Depot or Amazon and get some CRC 2-26 and wipe down the insides of the port with it (don't saturate it).
If it still doesn't work and you are under warranty or the second year promise, use it.
Otherwise you'll probably need to find someplace to replace the USB port.
Be really, really thorough on your cleaning. When this happened to me, I cleaned it several times - I THOUGHT I had done a good job - but nothing helped. I was ready to send it in, but the CRC 2-26 did the trick and it's been fine for a couple months now.