In addition to Ken Isakson's ...

In addition to Ken Isakson's comment, remember that the static level in the borehole or well is likely to be above the stream level, implying that the aquifer will be feeding the stream.  However when you pump, you will draw down the level in the well, which may result in a sufficient back-flow from the stream to the well.  So try to check both the static level without pumping, and again when you have been pumping for some time.

In terms of the chemical tests you could carry out, test the chloride in the stream and in the well, or even just the TDS.  If possible also try to get a sample to test from a well which is further from the stream but probably drawing from the same aquifer.  The results will indicate the approximate source of the water.  However as indicated above, try to get at least 2 samples from the well - i.e. after there has been no pumping for some hours, and then again after pumping has taken place for a period.

If the chloride or TDS values don't help you, try a non-conservative water quality parameter such as nitrates.