To separate contribution of the "ends" and the lateral surface, write $w=u+v$, where $u$ has zero boundary conditions on the lateral surface, and $v$ is zero on the ends.
The estimate $|u|\leq Ce^{-kt}\|\phi\|$ is true, where $k$ is the smallest eigenvalue of the Laplacan for $D$, and $t$ is the distance to the "ends" of the cylinder, and the constant $C$ depends on the norm used. This generalizes the similar estimate of Carleman for dimension 2.
Some references are:
H. KELLER, Sur la croissance des fonctions harmoniques s'annulant sur la frontiere d'un domaine non bornd. C. R. Acad. Sci. Paris 231 (1950), 266-267.
A. DINGHAS, Das Denjoy-Carlemansche Problem fur harmonische Funktionen in $E^n$. Det. Kgl. Norske Videns. Selsk. skr. (1962) No. 7, 12 pp.
A. HUBER, Ober Wachstumseigenschafien gewisser Klassen yon subharmonischen Funktionen. Comment. Math. Helv. 26 (1952), 81-116.
They all considered domains more general than cylinders, and worried about precise estimates. But for a straight cylinder, the stated estimate follows from very general compactness arguments, if you do not worry about the constant $C$. The key fact is that in the infinite cylinder ($D\times R$) all positive harmonic functions with zero boundary conditions are those with separated variables, so they have the form $u(x)(ae^{-kt}+be^{kt})$ where $x$ is the coordinate in $D$, and $u$ is a positive eigenfunction.