One of my concerns is the pins numbers on the DB9 on your schematic doesn't match the common pin2 and pin3 for transmit and receive. Maybe you numbered it differently, just a guess.
Use a scope / meter or some device to check each pin of your cable when it's plugged into your computer, but not your device. Send some chars from your computer, then determine which pin it shows up on the cable. Next plug the cable into your board. Repeat then walk the signal through your schematic. The characters from your comptuer should show up at the correct pins of U7 or you have a wiring problem.
Since I don't have a datasheet for part U7, I can't tell if it is wired correctly to the RS232 converter U8 chip. Do you have link to the datasheet for Q2686LOGIC U7 part???
Please revalidate and cross reference the pins of J18 to the RS232 cable. As I stated above, the pinout doesn't appear to match the common DB9 standard. I highly recommend that you number pins the same way on a schematic as they appear on the DB9 connector!!!! It is a BIG MISTAKE to number pins on a common connector in the wrong way, because it only causes confusion!!!
---------- Post added at 02:49 ---------- Previous post was at 02:36 ----------
The MAX3237E part has a working VCC voltage of 3.0V to 5.5V, but you are running it at 2.8V.