There are fixed prices for taxis booked inside the terminal (the safer taxis). When we visited, it was $11 to Miraflores but once we arrived, the driver kept going on about exchange rates and demanding more money. Resist! They have to abide by that price, which is already not so cheap compared to non airport rides.
Also ignore their suggestion of a different hotel. Stick to your original choice. They are only after a nice commission and it could mean you end up in an unsafe part of town.
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At Least Do This: It is indeed safer to travel from the airport in an official cab, booked in the terminal. Don't risk being mugged one hour after you arrive in Lima.
The cabs outside are cheaper but some will not be safe.
Alternative: Take the taxis but refuse to hand over more than the price that was displayed in the terminal. Refuse to go to their suggested hotel and demand they take you to your own choice of hotel.
If they are unpleasant, take their licence number and mention the tourist police, this should calm the driver down!