I had a story in mind that basically came from a true historical event in the 1930's. I won't be using real people for the characters but the main storyline is somewhat based on what actually happened. I'll try making it a little different, a little modern, but I think it'll be obvious where the idea about my storyline came from. Will that be considered rude or insensitive of me since actual people did die during those times in the past?
Historical fiction based on real events is a huge part of the genre of historical fiction. In fact, the taste today seems to be for stories that are as close to historical events as possible, with authors often basing their stories on one particular character (famous or otherwise) and often including notes almost apologizing when they rearrange actual events to make the story work more neatly. Historical fiction, in other words, is increasingly becoming documentary fiction.
I may be interpreting OPs intent incorrectly, but I think the other answers are a little off, as I don't think this is historical fiction but rather a modern retelling based on a historical event.
As an example, I see it as perhaps -- what if I had two characters like Bonnie and Clyde, but it is set in the 2000's. They are still a couple and go on a crime spree with similar events as to what actually happened and though their names are different, people who are familiar with Bonnie and Clyde would see the link.
Another way of viewing it, would be that if "Romeo and Juliet" had actually happened (as opposed to being a work of fiction itself), then OPs story would be "West Side Story".
And, yes this type of writing is quite common.
What you describe, if I understand it correctly, is historical fiction. That's a genre with a long and proud tradition. It includes works as diverse as Ivanhoe, War and Peace, The Three Musketeers, and All Quiet on the Western Front. It can be close to historical events, which, as Mark Baker points out, is the modern trend, or it can be as imaginative as Dumas' works. How true you stay to historic events is entirely up to you.
I would draw your attention in particular to Catch 22. It's hilarious (in parts). It's a satire. And it's about the soldiers in WW2. People died during those times. Quite a few of them, in fact. The tragedy doesn't mean you have to treat the time period like it's somehow sacred. You can write about it, people should write about it (otherwise, it would just be forgotten), you needn't be afraid to find the beautiful, and the funny, and the grotesque, in a hard situation.
If you go and imply that nobody in fact died in the ugly period you base your story on, or if you go "it's great that they died", people might get upset. Otherwise, have fun.
Yes, this is commonly called "Alternate History"
To cite from Wikipedia:
Alternate history or alternative history (Commonwealth English), sometimes abbreviated as AH, is a genre of fiction consisting of stories in which one or more historical events occur differently. These stories usually contain "what if" scenarios at crucial points in history and present outcomes other than those in the historical record. The stories are conjectural, but are sometimes based on fact. Alternate history has been seen as a subgenre of literary fiction, science fiction, or historical fiction; alternate history works may use tropes from any or all of these genres.
What you want to do is perfectly normal.
As long as you are not trying to make it a comedy and thereby being insensitive towards the groups of people that have died in that time you are free to write whatever comes to your mind.
Yes, people do this all the time. In science fiction, especially I think, there are lots of stories where someone took the fall of the Roman Empire and reworked it to be the fall of the Galactic Empire, or someone took Columbus discovery of America and reworked it into Earth's discovery of an alien planet, etc etc.
Yes, there are people who get offended when a work of fiction discusses some brutal reality of life, like murder or kidnapping or torture or rape. But if we're going to banish all things bad from fiction, what would we have left? Children's stories about Sally's first visit to the zoo, I guess. Even that some people would complain about the mistreatment of animals in zoos.
A good historical novel is well grounded in history, including "true historical events." The part that should be fictitious is that "your" characters are doing the heavy lifting.
You might have a major character that was a bodyguard to General George Washington. You should not have your character "lose" George Washington to the enemy (that would be historically inaccurate; a "near miss" is ok). Instead, the bodyguard character then uses his access to Washington for his own personal and professional development.
In most historical fiction, major characters in history should be minor characters in your novel, and major characters in your novel should be minor characters in history (as in the case of Washington and his bodyguard). The exception would be if your main character was a historical figure; George, or at least Martha Washington.