Masternode needs to be online. A good explanation from ChainCoin:

A Masternode is a server that performs service functions on a blockchain; such as instant send, coin mixing to support privacy, and aids in stabilizing the network. In return, masternodes are paid rewards, dividends in the form of Chaincoin on a periodic basis, for providing important services.


There are coins which supports rewards based on pure "holding" them in the wallet. (this is not staking) E.g. NEO (previously AntShares) is a specific example. For NEO, you would need to just keep coins in your wallet and you will be rewarded by NeoGAS (functions as Gas for Ethereum). This is more a dividend like approach.

In the case of NEO, you can keep your wallet completely offline and you're still getting rewards.