Let $H$ be a Hopf algebra and $K$ a Hopf subalgebra of $H$. If $H$ is finite-dimensional, then by the Nichols-Zoeller Theorem $H$ is free as a left (and right) module over $K$. 


Moreover, the same conclusion holds when $H$ is pointed, by the main theorem of 
 [D. Radford: Pointed Hopf algebras are free over Hopf subalgebras, J. Algebra 45, 266-273 (1977)], available here


<http://www.sciencedirect.com/science/article/pii/002186937790326X>


In both the previous cases, can one always find a basis $\mathcal{B}$ of $H$ over $K$ with $1\in \mathcal{B}$? 



Bases of free Hopf algebras over Hopf subalgebras
In the pointed case, this should follow from the proof in the Radford's paper available at the quoted link.


In your notation, consider the left $K$-module $C=KG$, where $G$ is the group of group-like elements of $H$. It is easy to see (and proved in the paper) that a basis of $C$ over $K$ is given by any set of right coset representatives for the subgroup of $G$ consisting of the group-like elements of $K$. 


Now, in order to prove that $H$ is free as left $K$-module Radford shows that there exists a certain series


$$
0=C^{(-1)}\subset C^{(0)} \subset C^{(1)} \cdots \subset C^{(\infty)}=\bigcup\_{n\geq -1}C^{(n)} =H
$$


with $C^{(0)}=C$ such that $C^{(n)}/C^{(n-1)}$ is a free left $K$-module for any $n\geq 0$.


How to find a basis with the desired property is then clear.


