Here is another translation
385. He for whom there is neither this shore nor the other shore, nor yet both, he who is free of cares and is unfettered — him do I call a holy man. 
Here's a footnote:
(v. 385) This shore: the six sense organs; the other shore: their corresponding objects; both: I-ness and my-ness.
Here's a Sutta which includes the parable of the 'raft' and 'other shore':
Asivisa Sutta: Vipers
"'The great expanse of water' stands for the fourfold flood: the flood
of sensuality, the flood of becoming, the flood of views, & the flood
'The near shore, dubious & risky' stands for self-identification. 'The
further shore, secure and free from risk' stands for Unbinding. 'The
raft' stands for just this noble eightfold path: right view, right
resolve, right speech, right action, right livelihood, right effort,
right mindfulness, right concentration. 'Making an effort with hands &
feet' stands for the arousing of persistence. 'Crossed over, having
gone to the other shore, he would stand on high ground, a brahman'
stands for the arahant."
I'm not sure why the Sutta praises "the further shore" whereas the Dhammapada praises non-awareness of shores. I suspect that the former is 'renunciation' whereas the latter is either 'middle way' or 'meditative absorption/concentration' ... but this/my interpretation might be non-standard i.e. wrong.
Here's a version in French (I don't know whether buddha-vacana.org has an English-language translation of the Dhammapada)
whose footnote gives the same explanation/interpretation of 'shore' as the accesstoinsight.org footnote does:
1 cette rive, l'autre rive: 'cette rive' correspond aux six āyatanas internes (œil, oreille, nez, langue corps, esprit), et 'l'autre rive' correspond aux six āyatanas externes (objets externes correspondant chacun des six āyatanas internes).
At the bottom of the page it says its translation is based on work by Daw Mya Tin and documents on the DLMB web site.
So, here (on the DLMB web site) is a detailed analysis of the Pali. It ends with a commentary which says,
Mara once approached the Buddha and asked him what does the word para (the other shore) mean. The Buddha knew that it was Mara and replied with this verse, saying that the other shore is the Nirvana, which only the Arahants can reach.
On this page (the work by Daw Mya Tin) is perhaps the simplest or most complete description. I copy-and-paste the whole below (but note that this web site has a similar page of description for each other verse in the Dhammapada):
Dhammapada Verse 385
Yassa param aparam va
paraparam na vijjati
tamaham brumi brahmanam.
Verse 385: Him I call a brahmana who has for him neither this shore (i.e., the sense-bases) nor the other shore (i.e., the sense objects), and who is undistressed and free from moral defilements1.
1 'This shore' and 'the other shore' are used in the sense of the internal and the external ayatanas. The internal ayatanas are the sense bases, viz, the eye, the ear, the nose, the tongue, the body and the mind; the external ayatanas are the sense objects, viz., visible object, sound, odour, taste, touch and mind-object.
For a true Brahmana (i.e., arahat) there is neither 'this shore' nor 'the other shore' which means that the senses of the arahat are calmed, and his passions extinguished.
The Story of Mara
While residing at the Jetavana monastery, the Buddha uttered Verse (385) of this book, with reference to Mara.
On one occasion, Mara came to the Buddha disguised as a man and asked him, "Venerable Sir, You often say the word 'param'. What is the meaning of that word?" The Buddha, knowing that it was Mara who was asking that question, chided him, "O wicked Mara! The words 'param' and 'aparam' have nothing to do with you. 'Param' which means 'the other shore' can be reached only by the arahats who are free from moral defilements."
Then the Buddha spoke in verse as follows:
Verse 385: Him I call a brahmana who has for him neither this shore (i.e., the sense-bases) nor the other shore (i.e., the sense objects ), and who is undistressed and free from moral defilements.
Given this translation I don't understand why the 'further shore' are 'sense-objects' and why that further shore can only be reached by the arahats. The internal/external ayatanas are introduced e.g. in this Wikipedia article.
Āyatana (Pāli; Sanskrit) is a Buddhist term that has been translated as "sense base", "sense-media" or "sense sphere." In Buddhism, there are six internal sense bases (Pali: ajjhattikāni āyatanāni; also known as, "organs", "gates", "doors", "powers" or "roots") and six external sense bases (bāhirāni āyatanāni or "sense objects"; also known as vishaya or "domains").
My only guess is that it might be related to some desire to 'see things as they really are, beyond illusion/subjectivity' or something like that.