This is literally - and I do not use that word lightly - the most insanely wrong climate-change-denial thought I have ever read
Let’s do some Actual Math (tm). The nearest star is about 4.4 light years, or 1.3 parsecs, away. That’s (rounding a little) about 280,000 times further from Earth than the sun.
(Technically it’s a triple star, but one of them is small and the other is megatiny so we will just count the sunlike one, Alpha Centauri A)
Amount of power you receive from something - it’s brightness - goes as the distance squared, so we get about 76 billion times less power from Alpha Centauri then the sun
That’s 0.00000001 watts per square meter. Rather not a lot, really.
But let’s imagine we move into a different, more exciting part of the Milky Way so our nearest neighbor is the most luminous giantest star in the galaxy. That’s about 10,000 times more luminous than the sun.
So we’d get an awe-inspiring 0.0001 extra watts per square meter. Terrifying! (Though it would light up the whole earth to suburban levels, so pretty irritating for astronomers.)
(And of course you’d only get that 0.0001 watts on the side facing the star, and Earth clouds would reflect most of it; averaged out it’s closer to 0.00001 watts per square meter.)
Compare that to about 1 watt per square meter of extra heat trapped by anthropogenic CO2. So even if we were right next to one of the biggest stars in the Milky Way, the effect on Earth’s temperature would be what we astronomers technically describe as “not much”.
As another example, the biggest cluster of stars in the milky way - Omega Centauri - has about ten million stars in a 150 light year diameter. If I put the Earth in the middle of it, most stars in it would still be a couple of million times further away than the sun.
So all told we’d get about (10 million)/(few million squared) suns = 0.001 watts per square meter. That’s about the same as being a hundred feet from a dim lightbulb, btw
Following up, it seems like the OP "clarified "that he was referring to a mangled press release lining up major glaciation periods with spiral arm crossings, based on earlier suggestions that cosmic rays can increase cloud formation.
The heritage of this is complicated; cosmic ray / cloud connections are usually cited in connection to the solar cycle as a climate change denial mechanism though the particular paper by Gies doesn't make that connection and is focused on the longer term.
Usually the cosmic ray / solar cycle / cloud thing is based on empirical correlations. The modern consensus seems to be (unsurprisingly) that this is not actually true - agupubs.onlinelibrary.wiley.com/doi/full/10.10… and journals.ametsoc.org/doi/10.1175/JC…
And of course even if there was a milky-way-arm-climate-correlation it would operate on timescales far far too slow to explain that past 100 years.
Overall this is one of those astronomers-should-stay-out-of-areas-they're-not-expert-in cautionary tales (possibly including me, as "interpreting what an original-climate-denier-actually-mean" is not my area of expertise.)
TL;DR: There's zero evidence for a mechanism in which our motion through the milky way could affect climate, and less than zero chance it has anything to do with what we've seen in the past 100 years, which is pretty much all on humans.
For example, here's a modern figure (from Agee et al) showing the lack of correlation in the recent solar cycle between cosmic ray excess ("Kiel") and cloud coverage (ISCCP). 
Also, more recent mapping of rotation of the galaxy shows that our passage through spiral arms doesn't line up with cold periods: iopscience.iop.org/article/10.108…
