Am I doing this counting problem correctly? $12$ people were asked which of the three television channel ch1, ch2, and ch3 they watch. How many possible results there could be? (For example, one result would be 8 people watch ch1, 10 people watch ch2, and no one watches ch3.

My answer is $12\times2^3$. Is this correct?

For each person, they could watch ch1 or not, then watch ch2 or not, and watch ch3 or not so for each person there are $2^3$ possibilities.
$12$ people are watching the television. A television can have $3$ states.

There are $3^{12} = 531441$ possibilities.

If you have zero or more non-viewers amongst these people the correct answer would be: $4^{12} = 16777216$ possibilities.

Description:

There is $c^p$ number of possibilities where $c$ equal the total number of channels and $p$ equal the total number of people watching.

Why?

no.  person 1   person 2  etc....
1       0          0
2       0          1
3       0          2
4       1          0
5       1          1
6       1          2
7       2          0
8       2          1
9       2          2
.       .          .
.       .          .


The truth table show that if there where only $2$ persons and only $3$ channels there would be $3^2 = 9$ possibilities. For larger number of people one could extend this truth table. Its increasing exponentially.
I think the answer should be :-

(1)If each person can choose only one channel :- (3)^12, as each person has three options, and each one could choose any one of the three.

(2) If a person has to choose atleast one channel :- (3C1 + 3C2 + 3C3)^12 = 7^12

(3) If a person can choose any number of channels (from 0 to 3) :- (3C0 + 3C1 + 3C2 + 3C3)^12 = 8^12