The effects will cancel out at some radius.
The star's gravitational potential is $\Phi(r)=-GM/r$ at distance $r$, where $M$ is the star's mass. If the star has radius $R$, the gravitational redshift at radius $r$ is $\Delta f/f\simeq \Phi(R)-\Phi(r)$ in the Newtonian limit. Here I adopt units in which the speed of light $c=1$. Meanwhile, the infalling observer's velocity at radius $r$ is $\sqrt{-2\Phi(r)}$, so the kinematic blueshift is $\Delta f/f\simeq \sqrt{-2\Phi(r)}$ in the Newtonian limit. The net frequency shift is thus zero when the two effects sum to zero, i.e. when
$$\Phi(R)-\Phi(r)+\sqrt{-2\Phi(r)}\simeq 0.$$
This simplifies to
$$\Phi(r)\simeq -\Phi(R)^2/2$$
at lowest order. In magnitude, the gravitational potential at the cancellation point is essentially the square of that at the star's surface. (And note that when $c=1$, gravitational potentials are dimensionless, so the dimensions here work out.)
The radius of the cancellation point is
$$r\simeq \frac{2R^2}{GM}=4\frac{R}{R_\mathrm{S}}R,$$
where $R_\mathrm{S}=2GM$ is the Schwarzschild radius associated with the star's mass. That is, the radius of the cancellation point is larger than the star's radius by about the same factor as the star's radius is larger than its Schwarzschild radius. For an ordinary star, that factor is large. (And if the star is not much larger than its Schwarzschild radius, then the Newtonian approximations I've made are not valid anyway.)
From the observer's point of view, if they do not detect any Doppler shift, then they might not detect they are moving toward the star and to their doom. How would the universe look to them at that moment, if they can only observe light and its frequency?
If the observer is precisely at the cancellation point, they will not detect any frequency shift. However, if they understand general relativity, then they will know that the starlight is gravitationally redshifted, and so they must be moving toward the star to cancel it.
Of course, this assumes perfect measurement capabilities. The gravitational potential at the surface of our Sun is of order $10^{-6}$, so the gravitational potential at the cancellation point for the gravitational and kinematic frequency shifts would be of order $10^{-12}$. Measuring frequency shifts of either of these magnitudes would be difficult.
PS. The expansion of the universe is ignored in the scenario, and only the star and the observer exists in this universe.
The expansion of the universe is irrelevant (not just negligible).
