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.