According to ***Draclvr's Windows 7 and Games***
thread, Stay Tuned for Danger was one of the Nancy Drews that wouldn't play correctly -- the sound wouldn't work. Judith, who tested, didn't say it wouldn't play at all, but the game may not be playable without sound. There are a lot of tweaks you can try, but it probably won't play correctly on Win7 outside of a Virtual Machine.
Install in your user's Documents folder instead of C:\Program Files or C:\Program Files(x86)
Install by right-clicking the Setup file on the game disc, selecting Compatibility tab, and changing Compatibility to Windows 95 or 98. Then check the box for "Run this program as an Administrator," save changes, and then try double-clicking the Setup to start the install.
Run the game by right-clicking the game shortcut, selecting Compatibility tab, change the dropdown to Windows 95 or 98 and check the boxes for disable visual themes, disable desktop composition, and lity to Windows 95 or 98. Then check the box for "Run this program as an Administrator." Save the changes and see what happens when you double-click the shortcut.
It's up to you if you think the game is worth all the hassle. If you want to play older games like this, I'd recommend locating an old computer -- maybe with a 200 to 400 MHz processor, with Windows 98. It's so much easier to set up and even stubborn games work.
You may also have to use a CPU slowdown program so certain timed parts of the game don't flash past like greased lightning. There was a patch to give you more time, but it's an old patch and doesn't increase your time enough to work well on newer computers. You'll probably need a save at that point -- assuming you're able to get the game to run. This isn't a Windows 7 problem -- it's a problem with running a game with a timer depending on CPU speed on a computer that's much faster than the computer the game was made for.
If you know someone who can set up Virtual PC or VMware on your computer, that's another option. However you'll probably still have difficulties with the timed parts.