Question: 

Can cookies compromise my security?

Answer: 

The quick answer is no, however they can compromise your privacy.

Cookies generally should not cause any alarm. There 2 types of cookies:

Trustworthy:

Those are cookies that keep track of your id. They can assist you in your e-commerce transactions. Also used for personalization, when you visit a site. I remember Amazon always greets me by name, when I visit the site and presents me with books relevant to what I have already purchased.

3rd party:

Used mainly by marketers and advertisers to track your browsing. Note: These cookies come from sites that advertise on sites you visit, therefore the name 3rd party.
You may wish to restrict 3 party cookies, if you wish to protect your privacy.

What you can do about it:

You can either delete your cookies manually, or tweak your browser for your desired setting.

In Firefox:

- Click on Tools and select Options

- Click on the padlock icon that says Privacy

Forefox here offers 3 condition for keeping cookies

1- they expire

This is the default option preference. When set, this allows websites to specify how long a cookie will be stored by Firefox. This makes it possible for a site to remember your preferences and login information
across browser sessions.

2- I close Firefox

With this option preference selected, Firefox will remove all stored cookies when closed.

3- ask me every time

With this option preference enabled, Firefox will ask you if you would like to keep a cookie (or refuse it entirely) every time a web site attempts to set one. This option could be annoying, but it’s worth you trying it in case you want to be in control of who is trying to insert a cookie file on your computer.

Regards.