is it safe to delete files in the download folder, since the files were saved someplace else on my computer. Will deleting & purging the download folder effect the way any of the programs work that I have downloaded?
From your description, you can probably delete the files in the Downloads directory without affecting anything. But your wording is a little ambiguous and there are a couple of caveats. For example, your question refers just to programs, but the Downloads directory is used for more than that. So to give you a safe answer, let me start with some context.
Since downloading is such a common action, the popular operating systems (you don't mention which one you use), include for convenience a directory called Downloads in the installed directory structure. It's set as the default download location so users can start using the computer without needing to configure that.
You can change it if you want, but the empty directory takes up no space, so there is really no need to delete the directory, itself.
The download directory receives all kinds of files--documents and media files, executables, software installation packages, etc. Those files remain there unless you move or delete them.
Documents, media files, and the like can be used regardless of where you put them. The only thing to be aware of regarding moving or copying them is re-accessing a file from the history (e.g., Recent Documents). If you open a file with your software while it is still in the Downloads directory and later move it somewhere else, the Recent Documents will still point to the old location and won't find it there.
Directly executable programs don't get installed, you just run them. You can do that in the download directory or move them somewhere else. To run them, you either go to the directory where they are, issue a command that includes the path to where they are, or create a shortcut of some kind that knows where the executable is located. If you move or copy an executable to somewhere else, any shortcut will need to be updated to point to where it is.
Software installation packages typically unpack the actual software files or download the ones you need, and store those elsewhere, creating system links to the working files. The installation package is then unnecessary. You can delete it, or save it in case you want to reinstall it without having to download it again. You can move or copy it; it won't make any difference where you run it from; it will always install its files where they are supposed to go.
Note, though, that a few packages will set up the working files wherever the installation package sits, especially if the instructions say to move the installation package to a destination directory before running it, or you are inattentive and miss a prompt asking where to install it. So verify that the download directory does not contain any working files or a child directory with working files. If it does, the easiest solution is to uninstall it and then reinstall it in the desired location.
Summary: Within the limitations described above (e.g., system links pointing to where the working files are stored, no inadvertent software installation in the download directory, and all files having copies elsewhere), the copies in the download directory are redundant and can be deleted without affecting anything. But there is no benefit to deleting the Downloads directory, itself.