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Unless you're using KeeperFX, try different Graphics modes using the DOSBox Configurator. You can load this from the shortcut "Launch Graphic Mode Setup".

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Have this problem too, will try to use different Graphics modes and then report back to you!

Changing the Graphic mode doesn't fix this problem in-game. Only OpenGL and OpenGL fix this problem for the main menu, but in-game then a black line appears above the game!

Have you tried switching resolution in-game using ALT+R?

Try this only when you are playing a level. It does not work on the menu.

That indeedly makes the game nicer, but it does not fix my problem. My problem is much worser: almost a quarter of the bottom of my screen is black! When I check "keep aspect ratio" at the advanced settings menu of Launch Graphic Mode Setup, it is fixed for a part (it looks like the image in post one then.)

I suggest editing the dosboxDK.conf with notepad and changing fullresolution=original to fullresolution=640x480

This should hopefully fix the big black bar problem. Let me know if this works for you.

No, this does not fix this issue. I'm using Windows 8.1 by the way. It now just looks like the image above.

I had a closer inspection of the screenshot posted by RevolutionSphere, but it doesn't appear that any graphics are missing.

The black border is there because the game resolution is 640x400, and DOSBox is probably trying to display it in 640x480. This means you have 80 pixels at the bottom of the screen which are black.

There is nothing else I can recommend aside from updating your DOSBox to the newest version.

You can very clearly see that the HUD on the left side of the screen is displaying fully.

This is a non-issue, really. Use KeeperFX if you want proper high resolution support.

This could be indeedly the case. Updating DOSBox doesn't fix this problem.

I might have to install KeeperFX then :\

Yes, this is the case. There is no other way I'm afraid.
Then do it! :P