It could be a faulty power button or it may be a faulty battery that is failing to hold its’ charge and that is why it won’t start when you press the power button.
Here’s a link to the user manual that shows how to open the toothbrush so that access to the NiMH rechargeable battery (and also the Power button) is possible. Scroll to the end of the document to see the procedure
They state that in doing this that the handle is destroyed BUT here’s a link that shows how to replace the battery in an Oral B 3709 toothbrush.
Since it has the same handle as a 3757 model (it says in 3757 manual that it has a 3709 handle) it should help to replace the battery in your model toothbrush as well.
Replacement NiMH batteries with solder tags are available at electronics parts stores.
It is inadvisable to try and solder standard NiMH batteries (that haven’t got the solder tag) as you have to apply too much heat and you can damage the battery and also injure yourself if it suddenly vents explosively.
If the battery is OK then the Power button (and/or its’ associated circuitry) needs to be tested to see if it is OK. If the button is faulty then replacing it may be problematical as it is doubtful that a replacement power button part is even available. I couldn’t find one perhaps you may have better luck.
Worth a try anyway.