The real problem is not that you just want to measure current. If so, you'd put the multimeter in current mode and connect the leads. You want to measure maximum current a source can safely deliver. That's very different.
The correct answer is you can't measure this, you have to get it from the datasheet. That is because at best you can only determine what your single sample does, which does not tell you what restrictions you have to design to to guarantee safe operation over the variation of possible units. You also can't always know what "safe" is for the device just from the outside.
However, for something with a normal digital output, we can make a few assumptions. Again, these are assumptions only and without a proper spec we can't know whether we are damaging the device. Let's say that we assume that dragging the voltage down to 90% of its open-circuit level will not hurt the digital output. You didn't say what voltage this port puts out nor provide any link, so we can only use hypothetical values. If this is a 5 V output, for example, then that means we assume we can load it in the high state without damage so that the voltage goes down to 4.5 V.
To measure the current the port is sourcing in this case, you first have to find the load that drags the output voltage to 4.5 V. Hook up the current meter in series with a variable resistor to the port when it is driving high. Use a separate meter to monitor the port voltage. Keep increasing the load (decreasing the resistance) until the voltage goes down to 4.5 V. The current meter is then showing you the current that the port can source at what we assumed was the safe voltage of 4.5 V.
If you don't have a suitable variable resistor, you can use a succession of fixed resistors to get close enough. Most likely the port will be able to source a mA or a few mA. A 4.5 kΩ resistor would draw 1 mA at 4.5 V. Start with 10 kΩ and go down from there carefully. You will probably end up with a resistor in the 500 Ω to 5 kΩ range.
If you use fixed known resistors, you can dispense with the current meter altogether. Start with 10 kΩ and keep decreasing the resistance connected to the port until its voltage gets to 4.5 V. Since you know the voltage and the resistance, the current the port is sourcing is simply that voltage divided by that resistance.