Let $f:X\rightarrow Y$ be a complex family of curves beetween singular surface $X$ and a smooth curve $Y,$ where $X$ has singularities on singular fibers of the family (rational double points as singularities). Let $S\subset Y$ be a discriminant locus of the family and $T=f^{*}S$ a normal crossing divisor. Let $D\subset X$ be a divisor on $X$ without vertical components. Is it possible to do in the same time process of semistability(fibers are reduced normal crossing divisors) of the family $f:X\rightarrow Y$ (by taking a ramified covering $Y'$ and then the desingularization $W$ of $X\times Y'$) obteining the family $f':W\rightarrow Y'$ and to assume that pullback of the divisor $T+D$ on $W$ is normal crossing divisor ?
I don't know what you mean by "the same time". Given any smooth surface any any divisor, we can always blow up enough times to make that divisor normal crossings. (I think this is discussed in Hartshorne?) So after disingularization, if you blow up a few more times, it will still be a designularization, but with $T+D$ normal crossings.
If we are are required to take the minimal disingularization of $X \times_Y Y'$, the answer is no. Consider a case when a fiber has one smooth component and one component of multiplicity 2. Then to get semistable reduction you must pull back by a covering of order a multiple of 2. Now $T$ should contain this fiber as it is singular. Let $D$ be a section intersecting the smooth locus. When we pull back by a ramified covering the intersection between teh omponent of $T$ and $D$ will no longer be a normal crossing.

$\begingroup$ in the same time I meant to get semistable family and to have the pullback of the divisor with normal crossings. $\endgroup$ – mne88 Oct 28 '16 at 13:47

$\begingroup$ Or, after semistability process if we do blowups to get the normal crossing divisor do we endenger the semistability of the family? $\endgroup$ – mne88 Oct 28 '16 at 13:50

$\begingroup$ @mne88 The blowups are just blowups of smooth points. So one just has to check that the blowup of a smooth point preserves semistability, which is clear. $\endgroup$ – Will Sawin Oct 28 '16 at 15:24

$\begingroup$ Do you have a reference for this last claim, that blow ups of smooth point preserves smistability? $\endgroup$ – mne88 Oct 29 '16 at 13:54

$\begingroup$ @mne88 Sorry this is wrong. The result I am thinking of only shows the fiber remains a simple crossings divisor  it does not necessarily remain reduced. $\endgroup$ – Will Sawin Oct 29 '16 at 14:15