(Xbox 360)
You have to open the game first.
Once in the main menu hit start then click on "add ons".
If you don't have the DLC it shouldn't display on that menu.
gaming.stackexchange.com/users/91398/zombieatmore
I don't know about console and pirated Skyrims, so I'm going to answer for the PC version:
As the game activates on steam, you can just right-click the game in your Steam library and choose to show DLC.
It's that simple.
gaming.stackexchange.com/users/101452/Izzo
Without opening Skyrim, you can go to %skyrimfolder%/Data and see if there is a file called "Dawnguard.esp".
If you open Skyrim and go to the data manager (the default one's name escapes me, but this will work with any mod manager)
you should also see that file as an option to be loaded.
The same is true for the other expansions.
Alternatively, if you are ingame, if you have Dawnguard the most obvious sign is that you will get a quest called "Dawnguard" as the opening breadcrumb quest, and hold guards will incessantly say "Heard they're reforming the Dawnguard!
Was thinking of joining up myself."
gaming.stackexchange.com/users/72049/IllusiveBrian
If you want to check to see if you have purchased it already, go to the Dawnguard page on Xbox.com.
If you are logged into the Microsoft Account tied to your GamerTag, it will say "Already Purchased" if you have bought it already.
You can also search the store from the console to see if you have bought it already.
To see if you have it downloaded, you can do as @deutschZuid suggested in the comments:
Go to Console Settings -> Storage, look for Skyrim in the list, and select it.
Dawnguard will be in the list if it is downloaded to your console.
gaming.stackexchange.com/users/461/Adeese
For an in-game solution, you can just travel to Fort Dawnguard and see if it's there.
You don't need to wait to be attacked by vampires or approached by the Dawngaurd to visit their fort.
I've created a new character and traveled directly there from Helgen before.
gaming.stackexchange.com/users/108003/DCShannon
I'm not sure if I do or do not have the Dawnguard DLC.  A lot of people say you have to be in Skyrim to see if you do; is there way I can find out if I have Dawnguard?
How do I tell if I have the Dawnguard DLC on Xbox 360?
gaming.stackexchange.com
<the-elder-scrolls-5-skyrim><xbox-360>
gaming.stackexchange.com/questions/210009
gaming.stackexchange.com/users/105659/Ethanlynch
As the game activates on steam, you can just right-click the game in your Steam library and choose to show DLC. You can also check if you purchased it, go to the Dawnguard page on Xbox.com. You can check if you own a specific DLC by many ways, such as checking the store page. It all comes down to whether you are using a console or PC.
Few ways that ownership of Dawnguard DLC can be checked.