Yes, the word Internet should be capitalized whenever it is used since it is a proper noun.
Whenever you refer to the Internet as a proper noun (ie you are referring to the World Wide Web), then Internet should be capitalized. If you are referring to a general network of computers, then you can use the lowercase internet. Most of the time, Internet will be capitalized.
internet or Internet
There is continual debate around whether the word Internet should continue to be capitalized since it has established itself into everyday life. In fact, the Uk has mostly transitioned to referring the to Internet with a lowercase “i” as you can see in this study done by Oxford Dictionaries. In the US, about 65% of the population refers to Internet with a capital “I” versus in the UK, only about 25% of the population uses a capital “I.” In fact, the earliest use of the word, cited in the Oxford English Dictionary from 1974, was with a lowercase i since at that time there were multiple internets.
While the debate continues, you should always follow the standard title capitalization rules for capitalizing titles regardless of the outcome of the Internet debate.