If the Euro continues to gain

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Good or bad for who? For some it's good, for others it's bad.

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Many currency traders out there look for any type of movement vs. the dollar whether it be strength or weakness. Higher interest rates would lead to a higher valued dollar therefore negating some of the effect on the strengthened Euro. This all depends on what market you are exposed to if you will be effected or not in a negative aspect. Currency day traders and swing traders look for any movement. It is the times when there is no defined strength where people lose money.

I hope this helped a bit.

It depends on factors like, where you live, and in which currency your income is denominated, and if you are an importer/exporter, or in a similar position to them.

For example, if you were in the EU, and earned your income by exporting stuff to the US, then a weak greenback (plus strong Euro) could reduce your effective income (in Euros.)

It depends on things like your own debt. Higher local interest rates would hurt you, if you had an adjustable rate mortgage, or other floating debt.

Also, it depends on your international exposure or involvement. For example, higher local interest, might just mean receiving more units of a lower-valued currency. Like getting more greenbacks from your investments, but each greenback is smaller than before.

If you are buying and/or selling in/from/to two different countries, then things get rather complicated, as far as what is good or bad with currency strengths.