Recursively add a file to all sub-directories

How about:

find . -type d -exec cp file {} \;

From man find:

   -type c
          File is of type c:
           d      directory

   -exec command ;
          Execute  command;  All following arguments to find are taken 
          to be arguments to the command until an  argument  consisting 
          of `;' is encountered.  The string `{}' is replaced by the 
          current file

So, the command above will find all directories and run cp file DIR_NAME/ on each of them.

If you just want to create an empty file, you can use touch and a shell glob. In zsh:

touch **/*(/e:REPLY+=/file:)

In bash:

shopt -s globstar
for d in **/*/; do touch -- "$d/file"; done

Portably, you can use find:

find . -type d -exec sh -c 'for d; do touch "$d/file"; done' _ {} +

Some find implementations, but not all, let you write find . -type d -exec touch {}/file \;

If you want to copy some reference content, then you'll have to call find in a loop. In zsh:

for d in **/*(/); do cp -p reference_file "$d/file"; done

In bash:

shopt -s globstar
for d in **/*/; do cp -p reference_file "$d/file"; done

Portably:

find . -type d -exec sh -c 'for d; do cp -p reference_file "$d/file"; done' _ {} +

When wanting to touch files called $name in the current directory and in all subdirectories, this will work:

find . -type d -exec touch {}/"${name}"  \;

Note that the comment by ChuckCottrill to the answer by terdon does NOT work, as it will only touch the file called $name in the current directory and the directory itself.

It will not create files in subdirectories as requested by the OP, while this version here will.