It seems very likely to me that every smooth connected $n$dimensional manifold with nonempty boundary has the homotopy type of a $(n1)$dimensional CW complex. Is that true and how to prove it? (or a counterexample?) Does this manifold need to be compact? What about $n$dimensional open manifolds? Thank you.

2$\begingroup$ Yes, see here for a combinatorial proof for any noncompact triangulated manifold. When $M$ is compact the proof is easier: the idea is to start with a triangulation and inductively "push in" cells with a free boundary face. $\endgroup$ – Mike Miller Dec 27 '18 at 21:37

2$\begingroup$ I wrote out a careful proof of this (based on the answers to my question that Mike Miller linked to) here: www3.nd.edu/~andyp/notes/NoncompactSurfaceFree.pdf $\endgroup$ – Andy Putman Dec 27 '18 at 23:04
I know of two proofs in the compact case. Let $M$ be a compact smooth $m$manifold with boundary $\partial M$.
1) Morse theory (Sketch). For this I think we need $m \ge 4$. There is a Morse function $f: M\to \Bbb [0,\infty)$ with $f^{1}(0) = \partial M$ and $f^{1}(1) = \emptyset$. One can assume $f$ is selfindexing. This will give a handlebody structure on $M$ relative to $\partial M$, i.e., $M$ is obtained from attaching handles of index at most $m$. By the technique of cancelling $0$handles, one can alter the handle decomposition so that there are no $0$handles. If we turn $f$ upside down, we obtain a handle decomposition of $M$ with no $m$handles. So $M$ is becomes a handlebody whose handles have index $< m$. This will imply (by an argument of Milnor) that $M$ has the homotopy type of a CW complex of dimension $\le m1$.
2) Wall's theory of finiteness ($m \ge 4$). First use that fact that $M$ is has the homotopy type of some CW complex (that is wellknown and can be proved in different ways). By Poincaré duakity, Wall's condition $D_{m1}$ is satisfied. Conseuqently, $M$ has the homotopy type of a CW complex of dimension $\le m1$ by Wall's "Theorem E." Wall's paper can be found here:
C.T.C. Wall, Finiteness Conditions for CWComplexes. Annals of Mathematics 81 (1965), pp. 56–69