The infrared properties of blazars can be studied from the statistical point of view with the help of sky surveys, like that provided by the Wide-field Infrared Survey Explorer and the Two Micron All Sky Survey. However, these sources are known for their strong and unpredictable variability, which can be monitored for a handful of objects only. In this paper, we consider the 28 blazars (14 BL Lac objects and 14 flat-spectrum radio quasars, FSRQs) that are regularly monitored by the GLAST-AGILE Support Program (GASP) of the Whole Earth Blazar Telescope since 2007. They show a variety of infrared colours, redshifts, and infrared-optical spectral energy distributions (SEDs), and thus represent an interesting mini-sample of bright blazars that can be investigated in more detail. We present near-IR light curves and colours obtained by the GASP from 2007 to 2013, and discuss the infrared-optical SEDs. These are analysed with the aim of understanding the interplay among different emission components. BL Lac SEDs are accounted for by synchrotron emission plus an important contribution from the host galaxy in the closest objects, and dust signatures in 3C 66A and Mrk 421. FSRQ SEDs require synchrotron emission with the addition of a quasar-like contribution, which includes radiation from a generally bright accretion disc (νLν up to ˜4 × 1046 erg s-1), broad-line region, and a relatively weak dust torus.