Context. The Ophiuchus clouds, in particular L 1688, are an excellent region to study the embedded phases of star formation, due to the relatively large number of protostars. However, the standard method of finding and characterizing embedded young stellar objects (YSOs) through just their infrared spectral slope does not yield a reliable sample. This may affect the age determinations, often derived from the statistics on the total number of embedded YSOs and pre-main sequence stars within a cloud.Aims. Our aim is to characterize the structure of protostellar envelopes on an individual basis and to correctly identify the embedded YSO population of L 1688.Methods. Spectral maps of the HCO⁺ j = 4-3 and C¹⁸O J = 3-2 lines, using the HARP-B array on the James Clerk Maxwell Telescope and SCUBA 850 μm dust maps, are obtained of all sources in the L 1688 region with infrared spectral slopes consistent with, or close to, that of embedded YSOs. Selected 350 μm maps obtained with the Caltech Submillimeter Observatory are presented as well. The properties, extent and variation of dense gas, column density and dust up to 1'(∼7500 AU) are probed at 15" resolution. Using the spatial variation of the gas and dust, together with the intensity of the HCOJ = 4-3 line, we are able to accurately identify the truly embedded YSOs and determine their properties. Results. The protostellar envelopes range from 0.05 to 0.5 M _⊙ in mass. The oncentration of HCO⁺ emission (∼0.5 to 0.9) is generally higher than that of the dust concentration. Combined with absolute intensities, HCO⁺ proves to be a better tracer of protostellar envelopes than dust, which can contain disk and cloud contributions. Our total sample of 45 sources, including all previously classified Class I sources, several flat-spectrum sources and some known disks, was re-classified using the molecular emission. Of these, only 17 sources are definitely embedded YSOs. Four of these embedded YSOs have little (0.1-0.2 M_⊙) envelope material remaining and are likely at the interesting transitional stage from embedded YSO to T Tauri star. About half of the flat-spectrum sources are found to be embedded YSOs and about half are disks.Conclusions. The presented classification method is successful in separating embedded YSOs from edge-on disks and confused sources. The total embedded population of the Ophiuchus L 1688 cloud is found almost exclusively in Oph-A, Oph-B2 and the Ophiuchus ridge with only three embedded YSOs not related to these regions. The detailed characterization presented will be necessary to interpret deep interferometric ALMA and future Herschel observations.