In my opinion there is no age limit in Kpop or Kdramas. People usually say ' you are too old to watch this' or 'you will get over it soon', but I never really understood this.
I think we shouldn't even label the music as Kpop. It's just music from a country different than the US. That doesn't mean that its worse or something that you will eventually stop listening to. Maybe you will stop listening to some artists, but that probably means that they don't express you anymore. Meaningful songs and talented people will always be listened to and I think that the same thing goes for 'Kpop'.
I think the same way about Kdrama too. There will probably be a time when some Kdrama won't satisfy you. For example if someone is watching only romance Kdrama, they may get bored of he genre that may seem less good after watching tons of romance dramas. Those people may stop watching Kdramas, because they are frustrated with the 'same old romance stories'. They will probably turn on other genres or they will completely stop watching Kdramas.
Kdramas are series. The right question is will you ever stop listening to music completely or stop watching movies and series?
Individuals believe that Kpop and Kdrama are childish and some of the songs or the series may be. That doesn't apply to everything though. There will be good songs and good series, but due to our personal taste changing, we might have to stop listening to songs that we used to, or genres that we loved.
In conclusion there is no age limit.
That is my opinion anyway :)