The usual solution is a MAX232 or similar IC.

The 8051 serial port is working with 0/5V (or 0/3V if Vcc is 3V) logic levels, the PC RS232 port with +/12V. To connect them, you need an interface cable, it inverts the signals and converts the levels. The MAX232 is a typical IC to do this job.

PC 9 pin RS232:
5 Gnd
2 RxD input
3 TxD output

It is possible to use a mobilephone "Data Cable", these cables are working with the above IC or similar. The older mobiles like the Ericsson 6/7/8xx are working with a 5V interface, the newer models are working with 3V.

I am using a ConCorde Ericsson 6/7/8xx cable, it is based on a Sipex SP385SPA IC.(the cable was about 5$, cheaper then the IC...)
Just cut the phone side connector and connect the proper wires to the 8051 board.

Another possibility: 2 npn transistors instead of the IC. Rather simple, the PC input, pin2 RxD is working with 0/5V levels too, so the first transistor is just for inverting the output of the 8051.

emitter: Gnd
base: 22k to the 8051_TxD(11)
collector: 10k to Vcc; wired to PC_RxD(2)

The other transistor is the board input.

emitter:Gnd
base: 22k to PC_TxD(3); 4.7k to Gnd
collector: 10 k to Vcc; wired to 8051_RxD(10)

The PC Gnd (5) connected to the 8051 board Gnd.