My instructor tells me that I must keep all my previous fingers dropped, for example, if I am to play a D on the A string in 1st position, I must not only drop my 3rd finger, but also my 1st and 2nd. He says that it is the standard that the Suzuki Method teaches. I am a bit confused by this as I see videos of classically trained violinists sometimes using one lower finger while the higher fingers are off of the fingerboard. What is better?
If you want to do vibrato on your fourth finger, you'll have a pretty hard time doing so while all other fingers are pressed down.
Doing position changes with all fingers down is also not going to be fun. Switching from 1st finger 1st position to 4th finger 3rd position while trying to lodge the 1st finger into its 3rd position location at the same time sounds downright awful.
So keeping all fingers down is not likely going to be your end game.
But how the intermediate steps are going to be organized by your "Method" is only loosely related. If you are hiking to a wedding, you don't start out wearing your tuxedo.
The answer depends on the music you are playing.
As a beginner it does no harm to keep all your fingers down, and may help you learn where the correct finger positions are to play in tune.
It is also wrong to "actively" lift the fingers a long way from the string and hold them in that position by tensing the muscles. You have to learn to just let them do whatever they want to do naturally when your hand is relaxed, apart from the finger(s) actually playing the notes.
When you start using vibrato, you will find that for the higher fingers (3 and 4) you don't want the lower fingers "pressed" onto the string, because they will restrict your freedom of movement.
On the other hand, in fast non-vibrato passages it is often a good idea to keep a lower finger firmly in contact with the string. For example if you are playing something like B D B E B D B E (first position on the A string) you don't want to keep lifting and lowering your first finger for every note B. Keep that finger on the string for the whole passage.
Yes, when you are beginning to learn the violin it is useful to keep all your fingers in position on the string when you play. You do this to help build muscle memory of the relative position of the fingers in the hand (how far apart fingers are from each other). It is important that you are not gripping or pressing with the fingers, just keeping them in contact with the string in the location of the notes you are using.
As you progress in your playing ability and can orient your fingers close enough to the notes to correct by ear you should progress to having the unused fingers relaxed, although still in roughly the position you will be playing them in: keeping the 2nd finger next to the first on the A string if you are going to play a C natural in the scale you are using, for example.
Vibrato training usually starts after you have started using your 4th finger, and are able to hear the notes in tune by ear. At that point, you will be using one finger at a time.
When learning to shift into positions it can be useful to use all fingers touching while in position for a while as well, as the fingers will be closer together as you go up in position.