How do stars from far away affect Earth?

A lot (to put it mildly) of elements are created in stars and supernovae. These elements then travel through space until they fall to Earth (or, to be exact, some microscopic portion of them reach us). Earth itself wouldn't exist if stars hadn't generated elements which then clumped into dust, into minerals, and so on until a big ball of matter started to orbit the Sun.

Here's a short quote from Wikipedia article on Cosmic ray:

Data from the Fermi space telescope (2013) has been interpreted as evidence that a significant fraction of primary cosmic rays originate from the supernovae of massive stars. However, this is not thought to be their only source. Active galactic nuclei probably also produce cosmic rays.

So I'll stand by my claim that stars are giving us mass (i.e. non-photons) as well as photons in real-time, not just as 5-billion-year-old space dust.

The stars in our galactic neighbourhood do have a dynamical, gravitational effect on the inner workings of the solar system:

They built the Oort cloud

The Oort cloud is a roughly spherical cloud of icy bodies that is thought to act as a reservoir of long-period comets (and which we speculate exists to explain said comets' existence). These icy bodies formed in much the same way that Kuiper belt objects did, accreting in circular orbits on the outer edges of the solar system, but then they had gravitational interactions with the planets that scattered them into higher orbits.

What happened next is probably best explained by Hal Levison in this interesting interview with Emily Lakdawalla:

You start out with a bunch of icy guys between the planets. And they start to scatter outward. If you think about it, you do planet flybys to get velocity kicks. Spacecraft do it all the time. The planet can't put you on an orbit that doesn't cross its orbit anymore, because it's just a velocity kick, it's not a position kick. So in the early stages, when the planets are forming, they're scattering these cometary objects around in such a way that the perihelion still remains among the planets, but the semimajor axis is doing a random walk as you get scattered. So the semimajor axis is bopping around and slowly growing because it's diffusing outward, until the point where the galaxy can become important gravitationally. And that acts as a torque. What the galaxy does is it doesn't change the semimajor axis, but it does change the perihelia distance. So you get out to a few thousand AU and the galaxy can lift the perihelia away from the planetary system and then you're frozen in the Oort cloud.

The orbits of the Oort cloud objects are huge because they were boosted there by gravity assists by the planets, but as Levison notes this process can only produce very tall ellipses that still intersect the planet's orbit - at the location of the original assist, because all elliptical orbits are closed.

The galaxy, on the other hand, is too far away to lift anyone's orbit up - it cannot provide orbital energy - but what it does is circularize these long, thin elliptical orbits. This is what Levison refers to as a torque: trading radial kinetic energy into angular velocity. This makes the orbit less elliptical while keeping the semimajor axis constant (because this keeps the energy constant), and it therefore pushes the perihelion away from the 'inner' solar system (i.e. from Neptune inward) until the object can no longer meaningfully interact with any planets, at which point the size of its orbit is sealed, and the only possible change that can happen is further drift of its ellipticity induced by extra-solar stars.

That feels like enough to be getting on with to me.

I don't think that light from the stars other than Sun is of much practical use nowadays except for the classic navigation, where it's essential of course.

I guess any effect comes from the limitless reach of the gravitational force, which drops with the square of the distance but grows linearly with the mass exerting the force. A star most obviously affects planets in its system. A bunch of stars are affecting their star cluster, and so on.

Check this subchapter in the Feynman's lectures.