If $B$ is a basis of order $k$ such that every integer $n$ can be written as a sum of $k$ elements from $B$ in $\asymp n^{o(1)}$ ways, then a simple counting argument yields $|B \cap [1 , X]| \asymp X^{\frac{1}{k}+o(1)}$. Thus a stronger estimate $|B \cap [1 , X]| \asymp (X \log X)^{\frac{1}{k}}$ in your problem is certainly a more interesting goal.

Theorem 8.6.3 in "The Probabilistic Method" by Alon & Spencer gives precisely a set $B$ satisfying this estimate when $k=3$ (and the proof can be adapted in order to handle any value $k \geq 3$).
They also give the following reference :

*Erdos, P. and Tetali, P. (1990). Representations of integers as the sum of k terms, 
Random Structures Algorithms 1(3): 245-261.*