Are there cognitive benefits to two hand typing versus one finger? I've been reading an old popular psychology book by russian psychiatrist Vladimir Levi. One of the things that he has his depressed patients try is learn to juggle. He insists that such activity brings both halves of the brain together and that the brain "likes to juggle". His book was written in the 1980s, and there's little science presented in the book to back the claim.

Juggling involves coordinating both hands and is automatic, if I understand correctly. Touch typing involves coordinating 10 fingers together and is also automatic once learned.

In the 1990s and early 2000s, computer keyboards were the dominant way of inputting data into a computer. If one was to type anything one used both hands together for extended periods of time (most likely touch typing with both hands). But as of 2013, more and more people report using tablet PCs to surf the web and interact with friends online. Such typing most likely involves one finger pressing the letters on a virtual keyboard. It's a much slower process, as the keyboard is too small for both hands. 

Does touch-typing has any cognitive benefits? Is the effect lost if one reverts to typing with a single finger?
Benefits of touch typing on task performance:
I assume that when comparing skilled individuals, two handed touch typing on a traditional keyboard is faster, more reliable, and more automatic, than the other methods of text input that you mention (e.g., phone or tablet keyboards). These advantages are discussed here.


Faster input means that ideas can be expressed more quickly.
Visual attention does not need to be allocated to the keyboard which permits more attention to be placed on the content or the screen.
The greater reliability and automaticity mean that less attention needs to be allocated to the task of typing than to other methods.
A more consistent keyboard layout across devices further justifies the investment in learning touch typing.


For further discussion of the cognitive and performance benefits of skilled touch typing, check out Yechiam et al (2003). Discussing the benefits of touch typing over visual typing, they state:


  One main distinction of the touch-typing strategy appears to be the
  ability to look at the screen while typing and to devote a minimal
  level of visual search to the keyboard (Cooper, 1983). This ability is
  gained through the memorization of key positions and finger
  trajectories, which makes touch-typing a difficult skill to acquire.
  Other differences between touch-typing and visually guided typing
  include touch typists' (a) use of all fingers of both hands, as
  opposed to the use of one hand of only some of the fingers; (b) fixed
  assignment of fingers to keys; (c) reduced arm movements; and (d)
  fixed locations of the palms (Crooks, 1964).


Broader cognitive benefits of touch typing:
While I have not read any specific research testing the idea, I think the benefits of touch typing relate to its superiority as a text entry method and not as some broader tool for brain integration. A large body of research on psychomotor and cognitive performance suggests that transfer across disparate domains tends to be minimal. The domains would need to have some overlapping elements. 

More broadly, there is a strong link between unemployment and depression (e.g., Frese &amp; Mohr, 1987). Engaging in meaningful or enjoyable activity might help improve mood in some cases (e.g., juggling, mastering some other new skill). However, I don't think the mechanism of that mood improvement would be hemispheric brain balancing. 

References


Cooper, W. E. (1983). Introduction. In W. E. Cooper (Ed.), Cognitive aspects of skilled typewriting (pp. 1-38). New York: Springer-Verlag.
Crooks, M. (1964). Touch typing for teachers. London: Pitman.
Frese, M., &amp; Mohr, G. (1987). Prolonged unemployment and depression in older workers: A longitudinal study of intervening variables. Social Science &amp; Medicine, 25(2), 173-178. PDF
Yechiam, E., Erev, I., Yehene, V., &amp; Gopher, D. (2003). Melioration and the transition from touch-typing training to everyday use. Human Factors: The Journal of the Human Factors and Ergonomics Society, 45(4), 671-684.