Preserving half of supersymmetries - why is this important?

The key idea is to recognize the stringy origin of the linear dilaton background as the near-horizon geometry of a stack of $NS5$ branes. You can check this "Brief review of little string theory" for technical details or Emergent Spacetime, sections 5.4.1 and 5.4.2 for intuitive arguments.

What does 'enough control over supersymmetry' mean? It simply means that you should preserve the sixteen spinor components you have shown as covariantly constant over the $X^{+}$ direction. When those components are preserved, you can identify the linear dilaton background with an $NS5$ brane solution. Interestingly, the $NS5$-brane solution (as seen in the low energy regime) is nonsingular, because of the “throat behavior” of its metric, but the dilaton grows without bound as one goes down the throat. Another interesting way to understand the throat behavior is to compactify a transverse direction to a stack of $k$ fivebranes and then apply a $T$-duality on it to obtain a Taub-NUT solution, such geometry can be taken to be an $A_{k-1}$ singularity (in the large raddi limit) and ultimately string theory is non-singular over those backgrounds (orbifolds).

How does supersymmetry help us negate the fact that as $X^{+}→−infty$ , the string coupling blows up? No, supersymmetry does not produce a weak coupling description of the theory of the linear dilaton at $X^{+}→−infty$. What actually does is to produce a concrete theory, that describe the strong coupling regime of the linear dilaton background, namely, the worldvolume theory of the $NS5$ brane or little string theory.

Despite of the absence of a microscopic theory that describe the quantum dynamics of Little String Theory, many things can be learned about that theory that are probably important to cosmology. Supersymmetry allow us to understand the DLCQ regularized description of the fivebrane within matrix theory (see TASI Lectures on Matrix Theory), its partition function (see On the partition sum of the NS five-brane) or even its Hagedorn behaviour at high temperatures that seems to be important to understand very early cosmologies in the context of string theory.

Just to be clear, supersymmetry does not prevent the blow up of the string coupling constant, but equip us with the hability to understand a precise theory that can be analized by different means in some regimes. Indeed, a (non-gravitational) holographic description for the dilaton background exist. Such construction is possible because the string interactions vanish at $X^{+} rightarrow infty $ and are indeed confined to the compact region at which $X^{+} rightarrow - infty $. The region outside the compact locii of the geometry where the theory is strongly coupled is called "the bulk theory" and the strongly coupled regime is identified as the holographic dual of the bulk theory, namely little string theory.