How would I design a show like CSI where only the organization is meant to fight off an alien invasion?
If it's set in the present, then your story will be limited by the technology available now. So full-on invasion battles wouldn't work in this scenario, because the aliens would have far superior technology (they travelled to Earth, after all), and those battles would be very short indeed if the aliens' goal is conquest. The story could work if set in the present, but it would have to be a series of discoveries, increasingly sinister, that point towards an alien invasion (or maybe they've been here all along?).
You'd have more latitude in terms of technology and the plot if you set it say, 300 years in the future. At that point, however, you'd also have to take into account how much society will have changed - it's entirely possible that paid labour will be a thing of the past, and all work will be handled by AI and robots.
About me tldr: Writer, editor, Cylon, and carnivore. Author of The Daywalker Chronicles. I self-published way before it was acceptable/ trendy to do so.
You can probably start with reviewing the XCom games.
If you can stomach oldie graphics and Nintendo Hard difficulty, you should start with the beginning, UFO: Enemy Unknown, released in 1994. Otherwise, you could start with the modern equivalent, XCOM: Enemy Unknown, released in 2012.
The modern one also has the expansion Enemy Within, which adds, among other things, EXALT, an enemy faction that seeks to embrace alien tech and outlast the invasion in order to rule the world afterwards.
To return to the question, the CSI-type show would probably focus on these games' Research labs, which do stuff ranging from alien autopsy to alien tech reverse engineering to alien interrogation.
How would I design a show like CSI where only the organization is meant to fight off an alien invasion? The problem you'll face is that in order to get to earth from wherever the aliens are from - light years away, they will need to have technology far more advanced than we have. So how would we successfully fight them? Nuclear weapons are very destructive, and they are also very visible - this would not be a secret or low key effort. Since nuclear weapons are so destructive, its better to have all this taking place outside of our atmosphere, so now you have bases on the moon, or other planets or scattered throughout our solar system.
Perhaps what the team is doing is identifying and destroying scouting expeditions and recon drones - very small forces.
You have re-written the show "UFO", of British origin. I think it was vintage 1975 or perhaps a bit later. Commander Ed Stryker was the main hero...
If you are planning to do so you can make the events in the year 2150 for example... when mankind discovered peaceful alien species and invited them to live and travel on our planet and together they created the CSI team you are talking about and worked together providing alien technology and the human brain to solve the crimes of aliens/humans that lives on Earth.