Which word should I use

This is a common confusion.

Minimum refers to "the least amount" whereas minimal means "a very little/small amount, often negligible."

Look at this difference:

The minimum contribution for this fund-raising is USD 10.
Jack contributed minimal amount. I certainly expected more as he's a businessperson.

Note that the minimal amount is your mind-set as you expected some great contribution. Even if Jack has contributed USD 1000, you had expected at least USD 5000 as he's a big businessperson. On the other hand minimum amount is the least maount (below that is not accepted).

In this sense, both the sentences are okay but to me, what are the minimum needs of your life sounds natural as we are talking about the least needs and not binding someone with a very negligible amount of something.

Context is everything! All you've done is provide four words, which is not nearly enough information to help you figure out which preposition would be best to use.

For example, someone could build a house on silt, so it tilted:

or, someone could build a house over silt, and it might tilt, too:

My pictures show the way I would differentiate between the two phrases. However, the prepositions are flexible enough in meaning so that you could still use over in the top picture, and on in the bottom picture, and I don't think I'd argue that those sentences would be "incorrect."

Macmillan defines over as:

in a higher position above someone or something, without touching them

(thus, the bottom house is over silt). However, Collins defines over as:

directly above; on the top of

so the first house is over silt as well.

These meanings aren't contradictory – they each simply list one (of many) possible meanings of a very multifaceted word. It's worth noting that Collins lists more than 10 defintions for over, and Macmillan lists around two dozen.