Do railroads on a different continent give the production bonus?

If your harbor is connected via railroad to your capital, the harbor connection counts as a railroad. You'll have then to lay railroads from your harbor on the other continent to any inland cities.

In the following example B,C and D would receive the bonus, E would not.

                   ~ ~ ~ ~ ~ ~ ~ 
|A|============|B|~ ~ ~ ~ ~ ~ ~ ~|C|===========|D|
                   ~ ~ ~ ~ ~ ~ ~  |
                                  `------------|E|

In the next example no city would receive the railroad bonus.

                   ~ ~ ~ ~ ~ ~ ~
|A|------------|B|~ ~ ~ ~ ~ ~ ~ ~|C|===========|D|
                   ~ ~ ~ ~ ~ ~ ~  |
                                  `------------|E|

As DMA57361 pointed out, if the capital has a harbor, any other city with a harbor automatically receives the railroad bonus. In the next example, C and D receive the bonus, E does not.

                   ~ ~ ~ ~ ~ ~ ~
               |A|~ ~ ~ ~ ~ ~ ~ ~|C|===========|D|
                   ~ ~ ~ ~ ~ ~ ~  |
                                  `------------|E|

Legend:

 A : capital
 = : railroad
 - : road
 ~ : ocean