I've begun downloading LaTeX and the download is 2GB. My MacBook Air has a 4GB 1600 Mhz DDR3 memory and a 1.3GHz Intel Core i5 processor. Will this large download have any impact of the long-term performance of my laptop considerably?
No, the speed differences should be unnoticeable as the data will be written to disk as it is downloaded. The only noticeable change may be in your internet speed as the download consumes bandwidth.
Adding application software will have no effect on the performance of your computer unless 1) the software is running or 2) it installs persistent background processes that consume resources. LaTeX does not do #2, so you need not worry that this will affect your computer in any negative way.