The WOLF video Grinitch linked to is worth watching.
Their main concern is that this ruling reifies 'transgender status' as a protected category under sex discrimination law, without defining 'transgender status'. Essentially the court has decided to create a new protected category that is based on someone's claim to be a sex that they aren't. It's not about dress codes, which are barely mentioned in the ruling.
They've introduced this idea that you can 'identify as' something that they acknowledge that you are not, which is just strange... Aimee Stephens just sent his employer a letter, and I guess this is sufficient to make the entire US government, every employer, every client, every business agree that someone is a sex, that everyone knows that they aren't. (25.20).*
Although the ruling specifies that this only applies to employment law, lower courts could decide that the reasoning is broadly applicable to other statutes. (Also, as Grinitch pointed out, employment law covers some instances where sex is relevant, e.g. rape crisis counsellor, prison guard, beauty therapist, professional athlete, where sex distinctions may now be illegal.)
This leaves the door open for lower courts to decide that, for instance, a male athlete who wants to compete against females has the right to do so if he claims 'transgender status'. To refuse him entry into the female category would be a form of sex discrimination against him. And so on for all areas of single-sex provision.
You can bet the ACLU will take this and run with it.
In the video, one of the WOLF lawyers also points out that even if SCOTUS eventually clarifies that single-sex provision is lawful under Title IX (and there's no guarantee they will), it often takes years, even a decade or longer, for a test case to reach the Supreme Court. In the meanwhile the lower courts are free to wreak havoc on single-sex provision, by interpreting it as a form of sex discrimination against persons who claim transgender status.
*Interestingly, I saw a conservative legal commentator chide the ADF (the group representing the employers in the Stephens case) for failing to challenge the ACLU's lawyers to define 'gender identity' during the arguments. He said in the future lawyers need to zero in on this point, because everything hinges on the fact that 'gender identity' has no objective definition.