Which group of an indepent variable is more likely to influence the dependent variable and by how much Give the following table
                  Male    |   Female   |   Total
Likes              10     |     3      |     13
Indifferent        20     |     20     |     40
Dislikes           10     |     9      |     19
Total              40           32           72

Here the independent variable is gender which is nominal. The dependent variable which is ordinal is either likers, indifferent, or dislikes. I can use the Chi Square test or Exact Fisher to determine if a H0 such as men and women are equally likely to be likers should be rejected.
However, the question I would like to answer after this is: which gender is significantly more likely than the other to be a liker? And, by how much?
For example: Men are x% significantly more likely than Women to be likers. 
I thought that the next step would be to perform a test between liker and each group: i.e. males and likers. But I got stumped because I thought I should be comparing males to females with respect to likers. 
Is there any standard way of approaching this question?
 A: 
I thought that the next step would be to perform a test between liker
  and each group: i.e. males and likers. But I got stumped because I
  thought I should be comparing males to females with respect to likers.

Chi-Square and generalized Fisher test do not address the ordinality of the second categorical variable.
To test if either gender is more prone to higher degree od liking (association between dichotomous and ordered categorical) I would suggest applying contingecny table trend test by Cochran and Armitage.
A: If you just want to say what proportion of men vs. women are likers (or something similar) you can do it straight from the table: 10/40 men are likers, 3/31 women are. 
You could then run a test of equality of proportions on these two proportions. 
If, on the other hand, you want to answer all the questions you asked in your text, and treat one variable as independent, then you want ordinal logistic regression. This would also allow you to control for any other variables.
However, your title question isn't well posed: With a 2-group independent variable, the influences are, by necessity, equal (and opposite). 
EDIT:
Here is how you would do it in SAS
data liking;
 input sex $ liking $ count;
 datalines;
 M 3 10
 F 3  3
 M 2 20
 F 2 20
 M 1 10
 F 1 9
;
run;

proc logistic data = liking;
 class sex;
 model liking = sex;
 freq count;
run;

and sex is not significant at p = 0.05. Its odds ratio is 1.69 with a 95% CI of 0.682 to 4.198.
