Older versions of Quicktime are often problematic in DOSBox. The usual solution is to replace the old version of Quicktime with version 2.1.2 which is known to work well in DOSBox. However, you have to wrestle with certain games, and such is the case with Critical Path.
Here are the steps I went through to get the game running:
Install Windows 3.11, install SB16 sound driver, install S3 video driver. I suggest using a video driver with 640x480 resolution because the game's UI is at that res; and the game supports 8-bit or 24-bit color, so choose 256 or 64K colors (I chose the latter).
Install Critical Path. The video and audio test will fail with errors, but ignore that.
Install Quicktime 2.1.2 and let it delete old version files, don't skip deleting any files.
So far it's all been pretty standard stuff; now for the bit of wrestling the game needs. The QT installer cleaned out all the old version files in \CRITPATH\QTW, but the game insists on MCIQTW.DRV being there, so copy that file into there from \WINDOWS\SYSTEM. Only that one file needs to be copied.
After that, the game seems to work fine, at least for the early portions of it that I played through.