Proper position of the right thumb for oboe, clarinet, and saxophone can affect player comfort, ease of technique, and risk of injury. (Thumb position is important for flute and bassoon as well, but I find the issues different enough that I won’t lump them in here.)
For oboe and clarinet, the right thumb supports the weight of the instrument. There’s a temptation to place the thumbrest near the base of the thumb, since it feels like a stronger, more stable position.
But this puts the fingers in a cramped and awkward position (upper joints removed for visibility):
Instead, the thumbrest should be positioned at or near the base of the thumbnail, which allows the right hand to be in a much more open and natural position.
If this feels too heavy, and especially if it causes pain or tension, the instrument can be supported with a neckstrap. This takes some of the weight off the thumb. Some oboe and clarinet thumbrests have rings for this purpose. There are also neckstraps available that can connect directly to the thumbrest itself, using a small leather tab.
For the saxophone, the main weight of the instrument is not supported by the thumb, but by a neckstrap. However, the right thumb should exert some forward pressure to help establish the correct angle between the mouthpiece and the embouchure. As with the oboe and the clarinet, the thumbrest should contact the thumb approximately at the base of the thumbnail to encourage a natural, relaxed position for the fingers.
As always, adaptations should be made as necessary to accommodate hands of varying shapes and sizes, but natural, unstrained position should be a priority.