As you are connecting only two devices together you can simplify things a bit by making a few alterations to your circuit.
>> see attached diagram <<
>> I picked this up on a website discussing design techniques - can't remember site name ! >>
The circuit I have (it works for me !) has a couple of features yours doesn't ...
... You seem to be missing the line resistors, one from 75176 pin 6 to +ve power and one from 75176 pin 5 to 0v (both 2k2), and you will require a resistor between pins 5 and 6 (about 100R) at one end of the line, possibly both.
... Your circuit is relying on the RS232 RTS line to control transmit/recieve.
This may or may not work as not all computers drive the RTS line correctly! M$ Windoze is reportedly very poor at timing, MSDOS is much better.
If you add the transistor the 75176 will be forced into transmit mode whenever data is sent on the RS232 line.
My circuit has recieve permanently enabled so the terminal program does not want local echo enabled, the interface (my version) performs an echo in hardware!
:? CAREFUL, you are using half-duplex, the two computers must not try to transmit at the same time, there is no detection of crashes.
I have the 75176 datasheet somewhere ... I just can't find it !