Comcast, and many other utility companies, will ask you for your SSN and will insist on it giving you many times with false answers.
I recommend you ask them if they have an alternative, which they most probably will say no because the person you are speaking with is not used to, and/or not trained to give alternatives.
The reason they ask for your SSN is for doing a credit check on you to make sure you are able to to pay your bills - and n case you don't - to ruin your life by ruining your credit score.
From their "Comcast Customer Privacy Notice":
What kind of personally identifiable information and CPNI does Comcast collect?
Comcast collects information from you at several different points when
you request, turn on, and use our services under an account we create
for you. Some of this information is personally identifiable
information, but much of it is not. We collect certain personally
identifiable information that our subscribers furnish to us in
connection with the provision of cable service or other services. In
order to provide reliable, high quality service to you, we keep
regular business records containing information about you that may
constitute personally identifiable information. These account records
include some, but typically not all, of the following information...
However, legally speaking, they cannot bind their services to your credit worthiness. They do provide an alternative when asked about it. Most probably they will ask some sort of a monetary deposit in order to open the account. This deposit may, or may not, be spread across several billing cycles. The deposit is their financial guarantee in case you fall behind on your payments. What should you do:
- Ask them what is the deposit amount.
- Can the deposit be spread across multiple bulling cycles
- What is the time period that that after it they will return you the deposit.