PRINT like THIS

GNU sed, 28

Score includes +2 for -rz passed to GNU sed.

s/(\S+\s*)(\S*)/\U\1\L\2/g

Test output

$ for t in 'Nineteen Eighty-Four (1984)' \
> 'Programming Puzzles & Code Golf   - 
> Hooray for Code Golf Stack Exchange!' \
> '     2be or not 2be, that is the ~~~QuEsTiOn~~~' \
> '++++++++[>++++[>++>+++>+++>+<<<<-]>+>+>->>+[<]<-]>>.>---.+++++++..+++.>>.<-.<.+++.------.--------.>>+.>++.'; do
> echo "$t" | sed -zrf PRINTlikeTHIS.sed
> done
NINETEEN eighty-four (1984)
PROGRAMMING puzzles & code GOLF   - 
HOORAY for CODE golf STACK exchange!
     2BE or NOT 2be, THAT is THE ~~~question~~~
++++++++[>++++[>++>+++>+++>+<<<<-]>+>+>->>+[<]<-]>>.>---.+++++++..+++.>>.<-.<.+++.------.--------.>>+.>++.
$ 

Vim script, 22 21 bytes

nm Q vEUWvEuWQ|norm Q

Explanation:

The first command maps Q to:

vEU make word uppercase
W go to next word
vEu make word lowercase
W go to next word
Q repeat recursive mapping

and |norm Q executes the mapping

Usage of the program

This is how you get Vim to read from stdin:

$ vim - -c 'nm Q vEUWvEuWQ|norm Q'
Vim: Reading from stdin
This is the first line!
This is the second line!
This is the third and last line I type to check if this works!
^D

This will open up Vim displaying this:

THIS is THE first LINE.
this IS the SECOND line.
THIS is THE third AND last LINE i TYPE to CHECK if THIS works!
~
~
~

Perl, 26 bytes

25 bytes code + 1 byte command line

s/\S+/++$i%2?uc$&:lc$&/eg

Thanks to ThisSuitIsBlackNot for fixing a couple of bugs!

Usage:

perl -p entry.pl