It depends where the alcohol came from, and the type of container it is in.
It is against Las Vegas municipal law to have an open glass or aluminum container of alcohol within 1,000 feet of a packaged liquor store. And at the Fremont Street Experience specifically, it is unlawful to have an open glass or aluminum container of any drink whether or not a package store is close by.
But carrying open containers of alcohol in paper or plastic is legal on the Fremont Street Experience only if the alcohol was bought at one of the large casinos such as the El Cortez, Golden Nugget, or Binion's. Alcohol that is bought at a bar with a "tavern-limited license" such as Insert Coin(s) and Vanguard has to stay on the premises.
People caught drinking from an open glass or aluminum container on Fremont Street or with an open container of alcohol from a "tavern-limited" bar face a citation that typically carries a $200 fine. Note that it is lawful to carry closed glass or aluminum containers of alcohol on Fremont Street as long as it is stapled closed in the package store bag with the receipt attached. For more information on Nevada "open container" laws, read our informational article on Nevada "open container" laws.