I contacted historian Bob Krick in Richmond and he was able to do some detective work with some materials in Richmond. According to Bob, Alexander Basham is buried at the Oakwood Cemetery in Section C. Unfortunately, we do not know exactly where in Section C he is buried. The graves are unmarked. They are numbered, but the corresponding names do not exist for many soldiers because over time the ladies who once maintained Oakwood could not read progressively fading markers, and some markers were lost, broken, or stolen. A 2nd list resurfaced about 15 years ago. This list was maintained by the caretaker of Oakwood Cemetery, John Redford. It's because of this list that we know for sure that Alexander Basham is buried at Oakwood Cemetery and in Section C (but not the specific plot).
As for the battle he was wounded in, Bob had the following to say (I've included the full correspondence):
"Attached is the scan of the appropriate page from the Oakwood Cemetery superintendent's burial list. That list only exists for 1861-62, so you are fortunate in that regard. The original list was discovered in the archives at the Museum of the Confederacy in Richmond a few years ago. Basham is about two-thirds of the way down the list, correct in every particular except it identifies his regiment as 6th Virginia instead of 60th Virginia. Those sorts of errors are typical, even routine, and it should not bother you.
I am happy to report that I finally found what I thought I remembered: a note in my files about a 60th Virginia casualty list. It was published in the Richmond Enquirer on July 7, 1862, page one. Happily for you, it breaks down the regimental casualties by battle: June 26 (Mechanicsville or Beaver Dam Creek); June 27 (Gaines's Mill); and June 30 (Frayser's Farm or Glendale). This is on microfilm, and I do not have the ability here in my office to make a copy for you from film.
It shows Alex Basham wounded "slightly in thigh" on June 30 at Frayser's Farm. So the inference is that he served in, and survived, the big charges at Mechanicsville and Gaines's Mill. You can guess along with me on how a "slight" thigh wound on June 30 became a fatal wound by the second week of July. Gangrene is one possibility, or some version of infection. Or, he could have been weakened by the wound and contracted something else.
The 60th Virginia's charge at Frayser's Farm is a very famous episode. The 60th and a sister regiment charged a battery of Union cannon and actually fought for possession of it using bayonets, fists, and clubbed muskets. The ground where that happened is now part of our national park, only recently preserved by the Civil War Trust and turned over to us. It is very, very different today. It is wooded, but was open in 1862. And it is unmarked today--no signs, no trails, etc. But in future years it slowly will become better looking, and better for visiting.
As a bonus, see the second attachment, which is a very well known piece of combat art depicting the 60th Virginia and its sister regiment the 55th Virginia charging that battery at Frayser's Farm. The artist served in that brigade, which makes the sketch that much better."