We describe recent spectroscopic observations of red giant stars made by the Space Telescope Imaging Spectrograph (STIS) instrument on board the Hubble Space Telescope, which has provided spatially resolved observations of the warm chromospheric winds that predominate for early K to mid-M giants. The H i Lyα lines of a set of 11 red giants observed with the STIS/E140M echelle grating are first analyzed to ascertain wind H i column densities and total wind mass-loss rates. The M giants have estimated mass-loss rates of
M
̇
=
(
14
–
86
)
×
10
−
11
M
⊙ yr−1, while the K giants with detected wind absorption have weaker winds with
M
̇
=
(
1.5
–
2.8
)
×
10
−
11
M
⊙ yr−1. We use long-slit spectra of H i Lyα for two particular red giants, α Tau (K5 III) and γ Cru (M3.5 III), to study the spatial extent of the Lyα emission. From these data we estimate limits for the extent of detectable emission, which are r = 193 R
* for γ Cru and r = 44 R
* for α Tau. The cross-dispersion emission profiles in the STIS echelle spectra of the larger sample of red giants also show evidence for spatial resolution, not only for H i Lyα but for other lines with visible wind absorption, such as Fe ii, Mg ii, Mg i, O i, and C ii. We characterize the nature of these spatial signatures. The spatial extent is far more apparent for the M giants than for the K giants, consistent with the stronger winds found for the M giants from the Lyα analysis.