A basis is a spanning set that is also independent. You are given that this subspace is spanned by these four vectors so a basis would be a subset of that. Start with any one of them, say, v1.

v2 is not a multiple of v1 so {v1, v2} are independent.

Now look at v3. Is v3 independent of v1 and v2? That is, can v3 be written as a linear combination of v1 and v2? That is the same as asking "can we have av1+ bv2+ cv3= 0 where a, b, c are not all 0?".

If yes, add v3. If no, ask the same about v4.

Since itself has dimension 3, a basis cannot contain more than 3 vectors.