From what I see in the datasheet of your PIR, it requires 5V (4.5 to 20) as power supply, but its logic output is 3.3V, which seems the same level of your BASYS board.
So, definately you can safely connect them provided the following:
Connect all the GNDs of every device together. As mentioned in another answer, two devices can communicate only when sharing a common reference.
Find a +5V somewhere to give power to your PIR. If the BASYS board has a power output of +5V, you can use that and you are done.
If your BASYS device does not have a +5V output, then you need another power supply. In this case, even 12V would do (the PIR accepts from 4.5 to 20 volt). The power supply has a GND pin and a +V pin. Again, connect the GND pin to the two GND pins of the two other devices. The +V pin goes in the only place it can go: the +PowerIN of the pir.
ANOTHER THING: when using logic (digital) levels, you can always put resistors in between. For example, you have to connect an output to an input? Well, connect them via a 1k resistor, it will work the same but it will not fry your devices in case of errors. Resistor values range normally from 100 ohm to even 10k, depending on the application, internal resistance (or impedence) of input and output, but 1k should always work. If it does not work, at least you does no damage.
FORGOT to say: probably a 3.3V output drives correctly a 5V input anyway - it is the contrary that can be problematic (but in this last case a resistor can do, even if it is not the most beautiful solution).