Putting support beam in basement
Put the end on the basement wall if you can. Is the basement wall poured concrete or block? If block, are the cores filled with grout?
If you are puting the beam under the joists you probably won't be able to place the beam on the wall (unless there is already a beam pocket there) and will need to support the end on a post. If so, I think you probably need to laterally support the post perhaps by bolting it into the wall.
Always make sure you have a good footing under every post.