Kinetic Inductance Detectors (KIDs) are superconductive low–temperature detectors useful for astrophysics and particle physics. We have developed arrays of lumped elements KIDs (LEKIDs) sensitive to microwave photons, optimized for the four horn–coupled focal planes of the OLIMPO balloon–borne telescope, working in the spectral bands centered at 150 GHz, 250 GHz, 350 GHz, and 460 GHz. This is aimed at measuring the spectrum of the Sunyaev–Zel'dovich effect for a number of galaxy clusters, and will validate LEKIDs technology in a space–like environment. Our detectors are optimized for an intermediate background level, due to the presence of residual atmosphere and room–temperature optical system and they operate at a temperature of 0.3 K. The LEKID planar superconducting circuits are designed to resonate between 100 and 600 MHz, and to match the impedance of the feeding waveguides; the measured quality factors of the resonators are in the 104 – 105 range, and they have been tuned to obtain the needed dynamic range. The readout electronics is composed of a cold part, which includes a low noise amplifier, a dc–block, coaxial cables, and power attenuators; and a room–temperature part, FPGA–based, including up and down-conversion microwave components (IQ modulator, IQ demodulator, amplifiers, bias tees, attenuators). In this contribution, we describe the optimization, fabrication, characterization and validation of the OLIMPO detector system.