What kinds of remote sensing required to observe methanogenic life on Titan?
Note: As I discovered after writing most of this answer, this question on Space Stack Exchange is closely related, and called2voyage’s answer uses some of the same early points that I do, as it discusses the paper you referenced.
It’s worth noting that the results of McKay &amp; Smith were purely theoretical, and their paper was originally received by the journal on the exact day the Huygens lander reached Titan: January 14, 2005. The paper was revised that April, but data from the spacecraft does not appear to have been published prior to that date. In other words, the authors had no idea what Cassini-Huygens would find on Titan. The empirical results confirmed their calculations; the “anomaly” you speak of had not yet been observed.
Later that year, Niemann et al. published results from the Gas Chromatograph Mass Spectrometer (GCMS). The results agreed with predictions of what Titan would look like with methanogenic life. Specifically, McKay &amp; Smith stated that there would be three main observable effects:

No acetylene (C2H2) on the surface
Very low levels of ethane (C2H6) compared to the expected amounts from photolysis. Photolysis on Titan has been studied by, among others, Romanzin et al. (2005), who include graphs of levels of ethane (and acetylene) at different altitudes.
A “sink” of molecular hydrogen at the surface, leading to a non-zero change in concentration per change in altitude
Niemann et al. found only trace amounts of acetylene, low levels of ethane. They provide a description of the GCMS (better summaries are in this presentation and Niemann et al. (1999)); I’ll give a quick overview here.

Gas from the atmosphere enters the gas sampling system, where it is first moved through a gas chromatographer (the carrier gas is hydrogen, chosen for storage efficiency), which separates gases using capillary columns. Gas is then sent through the ion source, which utilizes something called electron impact ionization, involving electron beams of 25 eV or 75 eV. A quadrupole switching lens can deflect the beams through mass filters; electron multipliers enhance detection in the low-mass regime.
The GCMS also found slightly lower levels of H2 on the surface than higher in the atmosphere; however, this difference was within the limits of uncertainty and so is not at all conclusive. That said, the Ion and Neutral Mass Spectrometer (INMS) onboard the Cassini orbiter did observe a gradient in the H2 mixing ratio. Strobel (2010) viewed this in the interpretation of the gradient supporting the methanogenic predictions of McKay &amp; Smith. The INMS, like the GCMS, uses an ion source and then directs the ions to a quadrupole mass analyzer; the two instruments are quite distinct, but they are still quite similar, so I’ll omit a description. A good overview can be found here.
The predictions contained in the paper you cited contained three predictions, all of which have been verified. It is true that they can be explained through abiotic phenomena, as the authors have pointed out. However, the Cassini-Huygens results mean that in order to conclusively prove that methanogenic life does not exist on Titan, other predictions would need to be falsified.
One prediction involves an azotosome, a hypothetical analogue of a cell membrane composed of acrylonitrile (created by Stevenson et al. (2015)). It is considered one of the best hopes for the construction of cells in a world based around methane and ethane, like Titan. Assuming the stability of azotosomes - which Stevenson et al. consider over about 100 years - one barrier for this type of methanogenic life on Titan would be a lack of atmospheric acrylonitrile. However, the INMS confirmed previous detections of acrylonitrile (see Nixon et al. (2010)). That said, concentrations would need to be high enough for life to be possible. More detailed mass spectroscopy on the surface, including samples from liquids, could answer this question.
Another potential piece of evidence which McKay &amp; Smith mentioned in passing is different levels of isotopes of carbon. Schulze-Makuch and Grinspoon (2005) note 12C/13C ratios of 95.6 found by the INMS, which would be caused by lifeforms processing carbon-based compounds, but Atreya et al. (2006) notes that this is only in the upper atmosphere; lower ratios (82.3, which is not drastically low but still too low) were found in the lower atmosphere. This would seem to be another strike against methanogenic life.
Finally, as McKay wrote in an article in 2010, there are abiotic explanations for the discrepancy in molecular hydrogen:

The Cassini-Huygens measurements are incorrect.
H2 particles are being physically moved down in the atmosphere.
There is a catalyst mediating a hydrogenation reaction. McKay writes that (referring to the existence of a catalyst operating at the low temperature of 95 Kelvin)

That would be quite interesting and a startling find although not as startling as the presence of life.



In summary, there are quite a few different factors that could influence whether or not methanogenic life can exist on Titan:

A lack of acetylene would indicate the presence of methanogenic life.
Low ethane levels could also be consistent.
A vertical gradient of molecular hydrogen would indicate a “sink”, possible biotic in origin.
The presence of high enough concentrations of acrylonitrile would indicate that life based on azotosomes is possible.
High carbon isotopes ratios could be positive indicators of methanogenic life.

Cassini-Huygens covered all of these to some extent using the Gas Chromatograph Mass Spectrometer and the Ion and Neutral Mass Spectrometer. Follow-up measurements to confirm these results, likely using mass spectroscopy, would shed more light on the possibility for methanogenic life, while new instruments designed to detect McKay’s alternative hypothetical catalyst could show that there is no need for life to explain the Cassini-Huygens measurements.