when you talk about mining and miner you insinuate by that POW (proof of work) consensus mechanism (a solution to the "Byzantine fault tolerance" problem). Other cryptos use for the same reason (reaching the consensus ) other mechanisms such as proof of stake (currently used in peercoin and Ethereum will switch to it soon) ,Proof of Activity, Proof of Burn, Proof of Capacity etc.

So every consensus algorithm or mechanism has it's own security model to secure the network and prevent double spending problem. POW depends on resolving a puzzle (mining: performing a certain amount of computational work) and an incentive the others has different procedure. Rather than that a proof of stake system requires the prover to show ownership of a certain amount of money.

For example in POS (proof f stake) the participants or the "provers" stake some of their coins so if they cheat they will lose their money (punish the cheater), to enforce the decentrlized consensus. In PoS cryptocurrencies the schema change: the blocks are usually said to be forged, or minted, rather than mined and there is no miner and no block's reward (only the transaction fees).

I suggest you to read more about POS, HERE.

the other consensus mechanism like Proof of Stake are not as safe as pure Proof of Work at an extent but every project is trying to provide a secure implementation.