The simple answer to this question is yes. It does matter how you hold your hands when you are playing the piano for a number of reasons. The most important reason is to prevent strain and injury and while I think that this is important for everyone who plays the piano, I think it is especially important for retirees and seniors because they can take longer to heal.
The best way to have your hands positioned over the piano keys, is to have your hands and fingers gently rounded. Your hands and fingers should feel very relaxed and comfortable. There should be no tension or stiffness in your hands at all. The part of the finger that should come into contact with the keys is either on the very tips of the fingers or as close to this as possible. Your fingers should not be flat at all. The heel of your hand should not be touching any part of the piano, yet so many times I have seen students try to rest the heel of their hand on the part just in front of the keyboard. Its also important to note that your forearm should be in a straight line with the back of your hand, so your wrist is not bent at all. This is the ideal.
Sometimes, when you are playing, the ideal is not possible, like when you are playing a few notes with the same hand and you need to stretch out your hand, of course it is going to need to flatten out somewhat. The ideal in this circumstance is to have your hand in the most comfortable position possible.
For more advanced piano players, sometimes it is unavoidable to use your thumb on a black note – this is acceptable when there is no other workable solution to playing something. If you do end up needing to use your thumb on a black note, do not try and twist your wrist to make the reach but rather slide your fingers up between the black notes so that your hands can stay in a comfortable position.
When a piece of piano music requires a section to be played loudly, then you can make the most of gravity by lifting your wrists and sometimes even forearms and shoulders so you can transfer weight and strength through your upper body to get more force (and therefore more volume) into the notes you are playing rather than trying to get a substantial force from your fingers alone.
When playing notes staccato, that is the notes are very short and detached and it is like your fingers are bouncing on the notes rather than playing the notes, it is not so much a finger movement that achieves this ‘bounce’ but rather it should be your wrists lightly flexing that will achieve the desired result. I have seen too many students achieve staccato by moving their forearms quickly. Doing it this way is more taxing on the arms and it doesn’t produce the best result, it can make these light staccato notes seem quite heavy and stodgy.
It does matter how you position your hands while playing the piano because it can protect you from strain and injury, keep your hands comfortable and relaxed whilst playing the piano. Correct hand positioning will also assist you in being able to achieve the best quality of sound.