The asker's own answer clarifies that a BenQ BL2420PT monitor was what wouldn't work with the passive adapters. That monitor has a native resolution of 2560x1440, which requires either DisplayPort, dual-link DVI, or HDMI 1.3+ (the monitor might still work with single-link DVI or HDMI 1.2− if the MacBook is set to output a resolution of 1920x1200 or lower):
The MacBook Air (Early 2014, both the 11-inch and 13-inch models) supports up to 2560x1600 over DisplayPort. So if the DisplayPort input on the monitor isn't already occupied, the cheapest and/or least confusing solution might be to use a Mini DisplayPort to DisplayPort cable.
Passive DisplayPort to dual-link DVI adapters do not exist due to DisplayPort's limited number of pins. There are active DisplayPort to dual-link DVI adapters available, and these require the use of a dual-link cable:
For HDMI, there are both active and passive adapters available which support HDMI 1.3+, however one vendor adapter (Plugable) claims:
When outputting a TMDS (DVI/HDMI) signal through Dual-Mode, most DisplayPort sources as well as the inexpensive passive adapters on the market are limited to a 165MHz clock rate. This limited clock rate generally limits the maximum resolution to 1920×1200 and reduces the maximum available color-depth to preserve the limited bandwidth.
Some mid-range DP -> HDMI adapters such as the Plugable MDPM-HDMIF are capable of operating at up to 300MHz, but ultimately they are still reliant upon the Dual-Mode functionality of the host to transmit the TMDS signal required for HDMI/DVI displays.
So your passive HDMI adapter might be one that is limited to 1920x1200. Nor is it clear from the many existing questions† on this topic whether the MacBook Air can output more than 1920x1200 over dual-mode DisplayPort.
† to name a few: Can a MacBook Air support 2560x1600 output via HDMI using a Mini DisplayPort/Thunderbolt to HMDI adaptor?, MacBookAir 2014 MiniDP to DP 2560x1440,
2013 Macbook Air not showing 1440p Option on External Display,
What's the max resolution for mini DisplayPort connected to a 2013 MacBook Air?, HDMI with resolution 2560 x 1440 possible?. Found using combinations of "macbook air", "displayport to hdmi", "passive", "dual mode" "2560x1400", "1440p", etc. Unfortunately many of these contain anecdotes, conflicting answers, or misleading advice (like forcing certain graphics or adapter hardware beyond specified limits, preferring HDMI over DisplayPort, etc.).