The problem is Beamdog updated BG:EE to version 2.6.6 but kept its dependencies as in versions 2.5.x. However in the meantime most GNU/Linux distributions upgraded said dependencies. So BG:EE 2.6.6 won't play in modern distributions because it expects openssl-1.0.x installed on your system (old library), while most distributions now have openssl-1.1.1.
The solution is simple, provided your distribution keeps older libs in their repositories.
(1) Find a file named openssl-1.0.x. in your distribution repositories. It is a compressed library, and the compression method depends on the distribution. Extract the files
libssl.so.1.0.0 and
libcrypto.so.1.0.0 to a new folder, say "old_libs_needed". Alternatively, if you have access to a computer running an older version of your distribution, locate those two files in /usr/lib64 (or usr/lib, depends on the distribution), and just copy them to your new folder.
(2) Move the folder "old_libs_needed" in your main BG:EE directory, where "start.sh" exists. This is a simple shell script that launches the game, and you have to modify it a little. Edit start.sh and add one line, so that it looks like that:
...
# Initialization
CURRENT_DIR="$( cd "$( dirname "${BASH_SOURCE[0]}" )" && pwd )"
export LD_LIBRARY_PATH=$LD_LIBRARY_PATH:$CURRENT_DIR/old_libs_needed cd "${CURRENT_DIR}"
....
(bold line is what you add in start.sh). Save and exit. The game should now run as usual.
For example if you are running Slackware 15.0 the old openssl you need can be found in Slackware 14.2 repositories (e.g.,
here), and it can be decompressed with the xz utility.
The same problem exists in BG2:EE and Icewind Dale:EE. It can be solved the exact same way. I don't have the other games mentioned above but if I were to bet, I'd say the solution should work there as well.
Now, I have nothing against Beamdog. In fact I believe the Enhanced Editions for the old Infinity games they released are great, and the best way to play those games nowadays. I also realize they can't prevent all kinds of problems caused by different versions of shared libraries. However in this case they could just include the old libraries needed in the game files.