Of course, in the context of Crisis itself, red skies and weird weather were the least of the Multiverse's worries. The first phase of Crisis (issues #1-4) involved waves of antimatter sweeping across all planes of existence, wiping out anything in their implacable paths. Issue #1 opened with an antimatter wave destroying the Crime Syndicate's home of Earth-Three, and by the start of issue #3 antimatter reached had reached Earth-One. The next phase of Crisis (issues #5-10) saw the five remaining Earths pulled by the Monitor's machine into a netherworld where they were safe from the antimatter waves, but welded together in an unhealthy symbiosis. Although two trips into the distant past (in issue #10) resulted in the creation of a single DC Universe, the heroes then had to fight off one last Anti-Monitor assault in issues #11-12. When this third phase began in issue #11, the red skies returned.
Since issues #11 and #12 showed the Earth pulled into the Anti-Matter Universe, and since the early issues featured the antimatter wave, perhaps the red skies were intended as a side effect of all that antimatter. However, the 20-years-later sequel Infinite Crisis kicked off with red skies across DC-Earth and no antimatter in sight. Therefore, it looked like the trope was moving away from specific meteorological consequences and into the realm of the merely creepy. Unless you're talking about sunrise or sunset, red skies are inherently unsettling, so in a sense it was inevitable that DC decouple them from any specific cosmic threat.
Besides, starting in 1992 the Batman animated series had been using red skies as part of its regular color palette. When combined with the black silhouette of Gotham City's skyline, it presented a stark backdrop to contrast with the show's array of colorful characters. The red skies didn't last throughout the various DC Animated Universe series, but since this period didn't have many Crisis-style comics crossovers to remind fans otherwise, the significance of red skies generally was diminished.